Triangle
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the basic geometric shape. For other uses, see Triangle (disambiguation).
Equilateral triangle
A regular triangle
Type
Regular polygon
Edges and vertices
3
Schläfli symbol
{3}
Coxeter diagram
Symmetry group
Dihedral (D3), order 2×3
Internal angle (degrees)
60°
Dual polygon
Self
Properties
Convex, cyclic, equilateral, isogonal, isotoxal
Triangle
A triangle
Edges and vertices
3
Schläfli symbol
{3} (for equilateral)
Area
various methods;
see below
Internal angle (degrees)
60° (for equilateral)
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C {\displaystyle \triangle ABC} .
In Euclidean geometry any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher dimensional Euclidean spaces this is no longer true. This article is about triangles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
Contents
[hide]
1 Types of triangle
1.1 By lengths of sides
1.2 By internal angles
2 Basic facts
2.1 Similarity and congruence
2.2 Right triangles
3 Existence of a triangle
3.1 Condition on the sides
3.2 Conditions on the angles
3.2.1 Trigonometric conditions
4 Points, lines, and circles associated with a triangle
5 Computing the sides and angles
5.1 Trigonometric ratios in right triangles
5.1.1 Sine, cosine and tangent
5.1.2 Inverse functions
5.2 Sine, cosine and tangent rules
5.3 Solution of triangles
6 Computing the area of a triangle
6.1 Using trigonometry
6.2 Using Heron's formula
6.3 Using vectors
6.4 Using coordinates
6.5 Using line integrals
6.6 Formulas resembling Heron's formula
6.7 Using Pick's theorem
6.8 Other area formulas
6.9 Upper bound on the area
6.10 Bisecting the area
7 Further formulas for general Euclidean triangles
7.1 Medians, angle bisectors, perpendicular side bisectors, and altitudes
7.2 Circumradius and inradius
7.3 Adjacent triangles
7.4 Centroid
7.5 Circumcenter, incenter, and orthocenter
7.6 Angles
8 Morley's trisector theorem
9 Figures inscribed in a triangle
9.1 Conics
9.2 Convex polygon
9.3 Hexagon
9.4 Squares
9.5 Triangles
10 Figures circumscribed about a triangle
11 Specifying the location of a point in a triangle
12 Non-planar triangles
13 Triangles in construction
14 See also
15 Notes
16 References
17 External links
Types of triangle
Euler diagram of types of triangles, using the definition that isosceles triangles have at least 2 equal sides, i.e. equilateral triangles are isosceles.
By lengths of sides
Triangles can be classified according to the lengths of their sides:
An equilateral triangle has all sides the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.[1]
An isosceles triangle has two sides of equal length.[note 1][2] An isosceles triangle also has two angles of the same measure, namely the angles opposite to the two sides of the same length; this fact is the content of the isosceles triangle theorem, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.[2] The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles.
A scalene triangle has all its sides of different lengths.[3] Equivalently, it has all angles of different measure.
Equilateral Isosceles Scalene
Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of "ticks", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, and a scalene triangle has different patterns on all sides since no sides are equal. Similarly, patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles since no angles are equal.
By internal angles
Triangles can also be classified according to their internal angles, measured here in degrees.
A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti[4] (singular: cathetus) of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, and 5 are a Pythagorean triple. The other one is an isosceles triangle that has 2 angles that each measure 45 degrees.
Triangles that do not have an angle measuring 90° are called oblique triangles.
A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, where a and b are the lengths of the other sides.
A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, where a and b are the lengths of the other sides.
A triangle with an interior angle of 180° (and collinear vertices) is degenerate.
A right degenerate triangle has collinear vertices, two of which are coincident.
A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral.
Right Obtuse Acute
⏟ {\displaystyle \underbrace {\qquad \qquad \qquad \qquad \qquad \qquad } _{}}
Oblique
Basic facts
A triangle, showing exterior angle d.
Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a 2-simplex (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC.
The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal).
The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.[5] This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.[note 2]
Similarity and congruence
Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.
Some basic theorems about similar triangles are:
If and only if one pair of internal angles of two triangles have the same measure as each other, and another pair also have the same measure as each other, the triangles are similar.
If and only if one pair of corresponding sides of two triangles are in the same proportion as are another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.[note 3]
Two triangles that are congruent have exactly the same size and shape:[note 4] all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)
Some individually necessary and sufficient conditions for a pair of triangles to be congruent are:
SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure.
ASA: Two interior angles and the included side in a triangle have the same measure and length, respectively, as those in the other triangle. (The included side for a pair of angles is the side that is common to them.)
SSS: Each side of a triangle has the same length as a corresponding side of the other triangle.
AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as AAcorrS and then includes ASA above.)
Some individually sufficient conditions are:
Hypotenuse-Leg (HL) Theorem: The hypotenuse and a leg in a right triangle have the same length as those in another right triangle. This is also called RHS (right-angle, hypotenuse, side).
Hypotenuse-Angle Theorem: The hypotenuse and an acute angle in one right triangle have the same length and measure, respectively, as those in the other right triangle. This is just a particular case of the AAS theorem.
An important condition is:
Side-Side-Angle (or Angle-Side-Side) condition: If two sides and a corresponding non-included angle of a triangle have the same length and measure, respectively, as those in another triangle, then this is not sufficient to prove congruence; but if the angle given is opposite to the longer side of the two sides, then the triangles are congruent. The Hypotenuse-Leg Theorem is a particular case of this criterion. The Side-Side-Angle condition does not by itself guarantee that the triangles are congruent because one triangle could be obtuse-angled and the other acute-angled.
Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.
Right triangles
The Pythagorean theorem
A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that
a 2 + b 2 = c 2 . {\displaystyle a^{2}+b^{2}=c^{2}.}
The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side c.
Some other facts about right triangles:
The acute angles of a right triangle are complementary.
a + b + 90 ∘ = 180 ∘ ⇒ a + b = 90 ∘ ⇒ a = 90 ∘ − b . {\displaystyle a+b+90^{\circ }=180^{\circ }\Rightarrow a+b=90^{\circ }\Rightarrow a=90^{\circ }-b.}
If the legs of a right triangle have the same length, then the angles opposite those legs have the same measure. Since these angles are complementary, it follows that each measures 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times √2.
In a right triangle with acute angles measuring 30 and 60 degrees, the hypotenuse is twice the length of the shorter side, and the longer side is equal to the length of the shorter side times √3:
c = 2 a {\displaystyle c=2a\,}
b = a × 3 . {\displaystyle b=a\times {\sqrt {3}}.}
For all triangles, angles and sides are related by the law of cosines and law of sines (also called the cosine rule and sine rule).
Existence of a triangle
Condition on the sides
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side. A triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality.
Conditions on the angles
Three given angles form a non-degenerate triangle (and indeed an infinitude of them) if and only if both of these conditions hold: (a) each of the angles is positive, and (b) the angles sum to 180°. If degenerate triangles are permitted, angles of 0° are permitted.
Trigonometric conditions
Three positive angles α, β, and γ, each of them less than 180°, are the angles of a triangle if and only if any one of the following conditions holds:
tan ⁡ α 2 tan ⁡ β 2 + tan ⁡ β 2 tan ⁡ γ 2 + tan ⁡ γ 2 tan ⁡ α 2 = 1 , {\displaystyle \tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}+\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}=1,} [6]
sin 2 ⁡ α 2 + sin 2 ⁡ β 2 + sin 2 ⁡ γ 2 + 2 sin ⁡ α 2 sin ⁡ β 2 sin ⁡ γ 2 = 1 , {\displaystyle \sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}+2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}=1,} [6]
sin ⁡ ( 2 α ) + sin ⁡ ( 2 β ) + sin ⁡ ( 2 γ ) = 4 sin ⁡ ( α ) sin ⁡ ( β ) sin ⁡ ( γ ) , {\displaystyle \sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )=4\sin(\alpha )\sin(\beta )\sin(\gamma ),}
cos 2 ⁡ α + cos 2 ⁡ β + cos 2 ⁡ γ + 2 cos ⁡ ( α ) cos ⁡ ( β ) cos ⁡ ( γ ) = 1 , {\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha )\cos(\beta )\cos(\gamma )=1,} [7]
tan ⁡ ( α ) + tan ⁡ ( β ) + tan ⁡ ( γ ) = tan ⁡ ( α ) tan ⁡ ( β ) tan ⁡ ( γ ) , {\displaystyle \tan(\alpha )+\tan(\beta )+\tan(\gamma )=\tan(\alpha )\tan(\beta )\tan(\gamma ),}
the last equality applying only if none of the angles is 90° (so the tangent function's value is always finite).
Points, lines, and circles associated with a triangle
There are thousands of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property: see the article Encyclopedia of Triangle Centers for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly encountered constructions are explained.
The circumcenter is the center of a circle passing through the three vertices of the triangle.
A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter, usually denoted by O; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle, called the circumdiameter, can be found from the law of sines stated above. The circumcircle's radius is called the circumradius.
Thales' theorem implies that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. If the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.
The intersection of the altitudes is the orthocenter.
An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute.
The intersection of the angle bisectors is the center of the incircle.
An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter, usually denoted by I, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.
The intersection of the medians is the centroid.
A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid or geometric barycenter, usually denoted by G. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its center of mass: the object can be balanced on its centroid in a uniform gravitational field. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side.
Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle.
The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.
Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).
The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
The center of the incircle is not in general located on Euler's line.
If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.
Computing the sides and angles
There are various standard methods for calculating the length of a side or the measure of an angle. Certain methods are suited to calculating values in a right-angled triangle; more complex methods may be required in other situations.
Trigonometric ratios in right triangles
Main article: Trigonometric functions
A right triangle always includes a 90° (π/2 radians) angle, here with label C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.
In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known as follows:
The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
The opposite side is the side opposite to the angle we are interested in, in this case a.
The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.
Sine, cosine and tangent
The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case
sin ⁡ A = opposite side hypotenuse = a h . {\displaystyle \sin A={\frac {\text{opposite side}}{\text{hypotenuse}}}={\frac {a}{h}}\,.}
This ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.
The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case
cos ⁡ A = adjacent side hypotenuse = b h . {\displaystyle \cos A={\frac {\text{adjacent side}}{\text{hypotenuse}}}={\frac {b}{h}}\,.}
The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case
tan ⁡ A = opposite side adjacent side = a b = sin ⁡ A cos ⁡ A . {\displaystyle \tan A={\frac {\text{opposite side}}{\text{adjacent side}}}={\frac {a}{b}}={\frac {\sin A}{\cos A}}\,.}
The acronym "SOH-CAH-TOA" is a useful mnemonic for these ratios.
Inverse functions
The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides.
Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse.
θ = arcsin ⁡ ( opposite side hypotenuse ) {\displaystyle \theta =\arcsin \left({\frac {\text{opposite side}}{\text{hypotenuse}}}\right)}
Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse.
θ = arccos ⁡ ( adjacent side hypotenuse ) {\displaystyle \theta =\arccos \left({\frac {\text{adjacent side}}{\text{hypotenuse}}}\right)}
Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side.
θ = arctan ⁡ ( opposite side adjacent side ) {\displaystyle \theta =\arctan \left({\frac {\text{opposite side}}{\text{adjacent side}}}\right)}
In introductory geometry and trigonometry courses, the notation sin−1, cos−1, etc., are often used in place of arcsin, arccos, etc. However, the arcsin, arccos, etc., notation is standard in higher mathematics where trigonometric functions are commonly raised to powers, as this avoids confusion between multiplicative inverse and compositional inverse.
Sine, cosine and tangent rules
Main articles: Law of sines, Law of cosines, and Law of tangents
A triangle with sides of length a, b and c and angles of α, β and γ respectively.
The law of sines, or sine rule,[8] states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is
a sin ⁡ α = b sin ⁡ β = c sin ⁡ γ . {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}.}
This ratio is equal to the diameter of the circumscribed circle of the given triangle. Another interpretation of this theorem is that every triangle with angles α, β and γ is similar to a triangle with side lengths equal to sin α, sin β and sin γ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sin α, sin β and sin γ. The side whose length is sin α is opposite to the angle whose measure is α, etc.
The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side.[8] As per the law:
For a triangle with length of sides a, b, c and angles of α, β, γ respectively, given two known lengths of a triangle a and b, and the angle between the two known sides γ (or the angle opposite to the unknown side c), to calculate the third side c, the following formula can be used:
c 2 = a 2 + b 2 − 2 a b cos ⁡ ( γ ) {\displaystyle c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )}
b 2 = a 2 + c 2 − 2 a c cos ⁡ ( β ) {\displaystyle b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )}
a 2 = b 2 + c 2 − 2 b c cos ⁡ ( α ) {\displaystyle a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )}
If the lengths of all three sides of any triangle are known the three angles can be calculated:
α = arccos ⁡ ( b 2 + c 2 − a 2 2 b c ) {\displaystyle \alpha =\arccos \left({\frac {b^{2}+c^{2}-a^{2}}{2bc}}\right)}
β = arccos ⁡ ( a 2 + c 2 − b 2 2 a c ) {\displaystyle \beta =\arccos \left({\frac {a^{2}+c^{2}-b^{2}}{2ac}}\right)}
γ = arccos ⁡ ( a 2 + b 2 − c 2 2 a b ) {\displaystyle \gamma =\arccos \left({\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)}
The law of tangents, or tangent rule, can be used to find a side or an angle when two sides and an angle or two angles and a side are known. It states that:[9]
a − b a + b = tan ⁡ [ 1 2 ( α − β ) ] tan ⁡ [ 1 2 ( α + β ) ] . {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}
Solution of triangles
Main article: Solution of triangles
"Solution of triangles" is the main trigonometric problem: to find missing characteristics of a triangle (three angles, the lengths of the three sides etc.) when at least three of these characteristics are given. The triangle can be located on a plane or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy, astronomy, construction, navigation etc.
Computing the area of a triangle
The area of a triangle can be demonstrated as half of the area of a parallelogram which has the same base length and height.
Calculating the area T of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is:
T = 1 2 b h {\displaystyle T={\frac {1}{2}}bh}
where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the side onto the line containing the side itself. In 499 CE Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.6).
Although simple, this formula is only useful if the height can be readily found, which is not always the case. For example, the surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.[10]
Using trigonometry
Applying trigonometry to find the altitude h.
The height of a triangle can be found through the application of trigonometry.
Knowing SAS: Using the labels in the image on the right, the altitude is h = a sin γ {\displaystyle \gamma } . Substituting this in the formula T = 1 2 b h {\displaystyle T={\frac {1}{2}}bh} derived above, the area of the triangle can be expressed as:
T = 1 2 a b sin ⁡ γ = 1 2 b c sin ⁡ α = 1 2 c a sin ⁡ β {\displaystyle T={\frac {1}{2}}ab\sin \gamma ={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta }
(where α is the interior angle at A, β is the interior angle at B, γ {\displaystyle \gamma } is the interior angle at C and c is the line AB).
Furthermore, since sin α = sin (π − α) = sin (β + γ {\displaystyle \gamma } ), and similarly for the other two angles:
T = 1 2 a b sin ⁡ ( α + β ) = 1 2 b c sin ⁡ ( β + γ ) = 1 2 c a sin ⁡ ( γ + α ) . {\displaystyle T={\frac {1}{2}}ab\sin(\alpha +\beta )={\frac {1}{2}}bc\sin(\beta +\gamma )={\frac {1}{2}}ca\sin(\gamma +\alpha ).}
Knowing AAS:
T = b 2 ( sin ⁡ α ) ( sin ⁡ ( α + β ) ) 2 sin ⁡ β , {\displaystyle T={\frac {b^{2}(\sin \alpha )(\sin(\alpha +\beta ))}{2\sin \beta }},}
and analogously if the known side is a or c.
Knowing ASA:[11]
T = a 2 2 ( cot ⁡ β + cot ⁡ γ ) = a 2 ( sin ⁡ β ) ( sin ⁡ γ ) 2 sin ⁡ ( β + γ ) , {\displaystyle T={\frac {a^{2}}{2(\cot \beta +\cot \gamma )}}={\frac {a^{2}(\sin \beta )(\sin \gamma )}{2\sin(\beta +\gamma )}},}
and analogously if the known side is b or c.
Using Heron's formula
The shape of the triangle is determined by the lengths of the sides. Therefore, the area can also be derived from the lengths of the sides. By Heron's formula:
T = s ( s − a ) ( s − b ) ( s − c ) {\displaystyle T={\sqrt {s(s-a)(s-b)(s-c)}}}
where s = a + b + c 2 {\displaystyle s={\tfrac {a+b+c}{2}}} is the semiperimeter, or half of the triangle's perimeter.
Three other equivalent ways of writing Heron's formula are
T = 1 4 ( a 2 + b 2 + c 2 ) 2 − 2 ( a 4 + b 4 + c 4 ) {\displaystyle T={\frac {1}{4}}{\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}}
T = 1 4 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) − ( a 4 + b 4 + c 4 ) {\displaystyle T={\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}
T = 1 4 ( a + b − c ) ( a − b + c ) ( − a + b + c ) ( a + b + c ) . {\displaystyle T={\frac {1}{4}}{\sqrt {(a+b-c)(a-b+c)(-a+b+c)(a+b+c)}}.}
Using vectors
The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors. Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then
| A B × A C | , {\displaystyle |\mathbf {AB} \times \mathbf {AC} |,}
which is the magnitude of the cross product of vectors AB and AC. The area of triangle ABC is half of this,
1 2 | A B × A C | . {\displaystyle {\frac {1}{2}}|\mathbf {AB} \times \mathbf {AC} |.}
The area of triangle ABC can also be expressed in terms of dot products as follows:
1 2 ( A B ⋅ A B ) ( A C ⋅ A C ) − ( A B ⋅ A C ) 2 = 1 2 | A B | 2 | A C | 2 − ( A B ⋅ A C ) 2 . {\displaystyle {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}.\,}
In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as:
1 2 | x 1 y 2 − x 2 y 1 | . {\displaystyle {\frac {1}{2}}\,|x_{1}y_{2}-x_{2}y_{1}|.\,}
Using coordinates
If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area can be computed as ​1⁄2 times the absolute value of the determinant
T = 1 2 | det ( x B x C y B y C ) | = 1 2 | x B y C − x C y B | . {\displaystyle T={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.}
For three general vertices, the equation is:
T = 1 2 | det ( x A x B x C y A y B y C 1 1 1 ) | = 1 2 | x A y B − x A y C + x B y C − x B y A + x C y A − x C y B | , {\displaystyle T={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{B}-x_{A}y_{C}+x_{B}y_{C}-x_{B}y_{A}+x_{C}y_{A}-x_{C}y_{B}{\big |},}
which can be written as
T = 1 2 | ( x A − x C ) ( y B − y A ) − ( x A − x B ) ( y C − y A ) | . {\displaystyle T={\frac {1}{2}}{\big |}(x_{A}-x_{C})(y_{B}-y_{A})-(x_{A}-x_{B})(y_{C}-y_{A}){\big |}.}
If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted.[12] The above formula is known as the shoelace formula or the surveyor's formula.
If we locate the vertices in the complex plane and denote them in counterclockwise sequence as a = xA + yAi, b = xB + yBi, and c = xC + yCi, and denote their complex conjugates as a ¯ {\displaystyle {\bar {a}}} , b ¯ {\displaystyle {\bar {b}}} , and c ¯ {\displaystyle {\bar {c}}} , then the formula
T = i 4 | a a ¯ 1 b b ¯ 1 c c ¯ 1 | {\displaystyle T={\frac {i}{4}}{\begin{vmatrix}a&{\bar {a}}&1\\b&{\bar {b}}&1\\c&{\bar {c}}&1\end{vmatrix}}}
is equivalent to the shoelace formula.
In three dimensions, the area of a general triangle A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC) is the Pythagorean sum of the areas of the respective projections on the three principal planes (i.e. x = 0, y = 0 and z = 0):
T = 1 2 | x A x B x C y A y B y C 1 1 1 | 2 + | y A y B y C z A z B z C 1 1 1 | 2 + | z A z B z C x A x B x C 1 1 1 | 2 . {\displaystyle T={\frac {1}{2}}{\sqrt {{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{vmatrix}}^{2}}}.}
Using line integrals
The area within any closed curve, such as a triangle, is given by the line integral around the curve of the algebraic or signed distance of a point on the curve from an arbitrary oriented straight line L. Points to the right of L as oriented are taken to be at negative distance from L, while the weight for the integral is taken to be the component of arc length parallel to L rather than arc length itself.
This method is well suited to computation of the area of an arbitrary polygon. Taking L to be the x-axis, the line integral between consecutive vertices (xi,yi) and (xi+1,yi+1) is given by the base times the mean height, namely (xi+1 − xi)(yi + yi+1)/2. The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal. The area of a triangle then falls out as the case of a polygon with three sides.
While the line integral method has in common with other coordinate-based methods the arbitrary choice of a coordinate system, unlike the others it makes no arbitrary choice of vertex of the triangle as origin or of side as base. Furthermore, the choice of coordinate system defined by L commits to only two degrees of freedom rather than the usual three, since the weight is a local distance (e.g. xi+1 − xi in the above) whence the method does not require choosing an axis normal to L.
When working in polar coordinates it is not necessary to convert to Cartesian coordinates to use line integration, since the line integral between consecutive vertices (ri,θi) and (ri+1,θi+1) of a polygon is given directly by riri+1sin(θi+1 − θi)/2. This is valid for all values of θ, with some decrease in numerical accuracy when |θ| is many orders of magnitude greater than π. With this formulation negative area indicates clockwise traversal, which should be kept in mind when mixing polar and cartesian coordinates. Just as the choice of y-axis (x = 0) is immaterial for line integration in cartesian coordinates, so is the choice of zero heading (θ = 0) immaterial here.
Formulas resembling Heron's formula
Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides a, b, and c respectively as ma, mb, and mc and their semi-sum (ma + mb + mc)/2 as σ, we have[13]
T = 4 3 σ ( σ − m a ) ( σ − m b ) ( σ − m c ) . {\displaystyle T={\frac {4}{3}}{\sqrt {\sigma (\sigma -m_{a})(\sigma -m_{b})(\sigma -m_{c})}}.}
Next, denoting the altitudes from sides a, b, and c respectively as ha, hb, and hc, and denoting the semi-sum of the reciprocals of the altitudes as H = ( h a − 1 + h b − 1 + h c − 1 ) / 2 {\displaystyle H=(h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1})/2} we have[14]
T − 1 = 4 H ( H − h a − 1 ) ( H − h b − 1 ) ( H − h c − 1 ) . {\displaystyle T^{-1}=4{\sqrt {H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}}.}
And denoting the semi-sum of the angles' sines as S = [(sin α) + (sin β) + (sin γ)]/2, we have[15]
T = D 2 S ( S − sin ⁡ α ) ( S − sin ⁡ β ) ( S − sin ⁡ γ ) {\displaystyle T=D^{2}{\sqrt {S(S-\sin \alpha )(S-\sin \beta )(S-\sin \gamma )}}}
where D is the diameter of the circumcircle: D = a sin ⁡ α = b sin ⁡ β = c sin ⁡ γ . {\displaystyle D={\tfrac {a}{\sin \alpha }}={\tfrac {b}{\sin \beta }}={\tfrac {c}{\sin \gamma }}.}
Using Pick's theorem
See Pick's theorem for a technique for finding the area of any arbitrary lattice polygon (one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points).
The theorem states:
T = I + 1 2 B − 1 {\displaystyle T=I+{\frac {1}{2}}B-1}
where I {\displaystyle I} is the number of internal lattice points and B is the number of lattice points lying on the border of the polygon.
Other area formulas
Numerous other area formulas exist, such as
T = r ⋅ s , {\displaystyle T=r\cdot s,}
where r is the inradius, and s is the semiperimeter (in fact this formula holds for all tangential polygons), and[16]:Lemma 2
T = r a ( s − a ) = r b ( s − b ) = r c ( s − c ) {\displaystyle T=r_{a}(s-a)=r_{b}(s-b)=r_{c}(s-c)}
where r a , r b , r c {\displaystyle r_{a},\,r_{b},\,r_{c}} are the radii of the excircles tangent to sides a, b, c respectively.
We also have
T = 1 2 D 2 ( sin ⁡ α ) ( sin ⁡ β ) ( sin ⁡ γ ) {\displaystyle T={\frac {1}{2}}D^{2}(\sin \alpha )(\sin \beta )(\sin \gamma )}
and[17]
T = a b c 2 D = a b c 4 R {\displaystyle T={\frac {abc}{2D}}={\frac {abc}{4R}}}
for circumdiameter D; and[18]
T = tan ⁡ α 4 ( b 2 + c 2 − a 2 ) {\displaystyle T={\frac {\tan \alpha }{4}}(b^{2}+c^{2}-a^{2})}
for angle α ≠ 90°.
The area can also be expressed as[19]
T = r r a r b r c . {\displaystyle T={\sqrt {rr_{a}r_{b}r_{c}}}.}
In 1885, Baker[20] gave a collection of over a hundred distinct area formulas for the triangle. These include:
T = 1 2 [ a b c h a h b h c ] 1 / 3 , {\displaystyle T={\frac {1}{2}}[abch_{a}h_{b}h_{c}]^{1/3},}
T = 1 2 a b h a h b , {\displaystyle T={\frac {1}{2}}{\sqrt {abh_{a}h_{b}}},}
T = a + b 2 ( h a − 1 + h b − 1 ) , {\displaystyle T={\frac {a+b}{2(h_{a}^{-1}+h_{b}^{-1})}},}
T = R h b h c a {\displaystyle T={\frac {Rh_{b}h_{c}}{a}}}
for circumradius (radius of the circumcircle) R, and
T = h a h b 2 sin ⁡ γ . {\displaystyle T={\frac {h_{a}h_{b}}{2\sin \gamma }}.}
Upper bound on the area
The area T of any triangle with perimeter p satisfies
T ≤ p 2 12 3 , {\displaystyle T\leq {\tfrac {p^{2}}{12{\sqrt {3}}}},}
with equality holding if and only if the triangle is equilateral.[21][22]:657
Other upper bounds on the area T are given by[23]:p.290
4 3 T ≤ a 2 + b 2 + c 2 {\displaystyle 4{\sqrt {3}}T\leq a^{2}+b^{2}+c^{2}}
and
4 3 T ≤ 9 a b c a + b + c , {\displaystyle 4{\sqrt {3}}T\leq {\frac {9abc}{a+b+c}},}
both again holding if and only if the triangle is equilateral.
Bisecting the area
There are infinitely many lines that bisect the area of a triangle.[24] Three of them are the medians, which are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter. There can be one, two, or three of these for any given triangle.
Further formulas for general Euclidean triangles
See also: List of triangle inequalities
The formulas in this section are true for all Euclidean triangles.
Medians, angle bisectors, perpendicular side bisectors, and altitudes
Main articles: Median (geometry), Angle bisector, Bisection § Perpendicular bisectors, and Altitude (geometry)
The medians and the sides are related by[25]:p.70
3 4 ( a 2 + b 2 + c 2 ) = m a 2 + m b 2 + m c 2 {\displaystyle {\frac {3}{4}}(a^{2}+b^{2}+c^{2})=m_{a}^{2}+m_{b}^{2}+m_{c}^{2}}
and
m a = 1 2 2 b 2 + 2 c 2 − a 2 = 1 2 ( a 2 + b 2 + c 2 ) − 3 4 a 2 {\displaystyle m_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\sqrt {{\frac {1}{2}}(a^{2}+b^{2}+c^{2})-{\frac {3}{4}}a^{2}}}} ,
and equivalently for mb and mc.
For angle A opposite side a, the length of the internal angle bisector is given by[26]
w A = 2 b c s ( s − a ) b + c = b c [ 1 − a 2 ( b + c ) 2 ] = 2 b c b + c cos ⁡ A 2 , {\displaystyle w_{A}={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}={\sqrt {bc\left[1-{\frac {a^{2}}{(b+c)^{2}}}\right]}}={\frac {2bc}{b+c}}\cos {\frac {A}{2}},}
for semiperimeter s, where the bisector length is measured from the vertex to where it meets the opposite side.
The interior perpendicular bisectors are given by
p a = 2 a T a 2 + b 2 − c 2 , {\displaystyle p_{a}={\frac {2aT}{a^{2}+b^{2}-c^{2}}},}
p b = 2 b T a 2 + b 2 − c 2 , {\displaystyle p_{b}={\frac {2bT}{a^{2}+b^{2}-c^{2}}},}
p c = 2 c T a 2 − b 2 + c 2 , {\displaystyle p_{c}={\frac {2cT}{a^{2}-b^{2}+c^{2}}},}
where the sides are a ≥ b ≥ c {\displaystyle a\geq b\geq c} and the area is T . {\displaystyle T.} [27]:Thm 2
The altitude from, for example, the side of length a is
h a = 2 T a . {\displaystyle h_{a}={\frac {2T}{a}}.}
Circumradius and inradius
Main articles: Circumradius and Inradius
The following formulas involve the circumradius R and the inradius r:
R = a 2 b 2 c 2 ( a + b + c ) ( − a + b + c ) ( a − b + c ) ( a + b − c ) ; {\displaystyle R={\sqrt {\frac {a^{2}b^{2}c^{2}}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}};}
r = ( − a + b + c ) ( a − b + c ) ( a + b − c ) 4 ( a + b + c ) ; {\displaystyle r={\sqrt {\frac {(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)}}};}
1 r = 1 h a + 1 h b + 1 h c {\displaystyle {\frac {1}{r}}={\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}}
where ha etc. are the altitudes to the subscripted sides;[25]:p.79
r R = 4 T 2 s a b c = cos ⁡ α + cos ⁡ β + cos ⁡ γ − 1 ; {\displaystyle {\frac {r}{R}}={\frac {4T^{2}}{sabc}}=\cos \alpha +\cos \beta +\cos \gamma -1;} [7]
and
2 R r = a b c a + b + c {\displaystyle 2Rr={\frac {abc}{a+b+c}}} .
The product of two sides of a triangle equals the altitude to the third side times the diameter D of the circumcircle:[25]:p.64
a b = h c D , b c = h a D , c a = h b D . {\displaystyle ab=h_{c}D,\quad \quad bc=h_{a}D,\quad ca=h_{b}D.}
Adjacent triangles
Suppose two adjacent but non-overlapping triangles share the same side of length f and share the same circumcircle, so that the side of length f is a chord of the circumcircle and the triangles have side lengths (a, b, f) and (c, d, f), with the two triangles together forming a cyclic quadrilateral with side lengths in sequence (a, b, c, d). Then[28]:84
f 2 = ( a c + b d ) ( a d + b c ) ( a b + c d ) . {\displaystyle f^{2}={\frac {(ac+bd)(ad+bc)}{(ab+cd)}}.\,}
Centroid
Main article: Centroid
Let G be the centroid of a triangle with vertices A, B, and C, and let P be any interior point. Then the distances between the points are related by[28]:174
( P A ) 2 + ( P B ) 2 + ( P C ) 2 = ( G A ) 2 + ( G B ) 2 + ( G C ) 2 + 3 ( P G ) 2 . {\displaystyle (PA)^{2}+(PB)^{2}+(PC)^{2}=(GA)^{2}+(GB)^{2}+(GC)^{2}+3(PG)^{2}.\,}
The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:
A B 2 + B C 2 + C A 2 = 3 ( G A 2 + G B 2 + G C 2 ) . {\displaystyle AB^{2}+BC^{2}+CA^{2}=3(GA^{2}+GB^{2}+GC^{2}).} [29]
Let qa, qb, and qc be the distances from the centroid to the sides of lengths a, b, and c. Then[28]:173
q a q b = b a , q b q c = c b , q a q c = c a {\displaystyle {\frac {q_{a}}{q_{b}}}={\frac {b}{a}},\quad \quad {\frac {q_{b}}{q_{c}}}={\frac {c}{b}},\quad \quad {\frac {q_{a}}{q_{c}}}={\frac {c}{a}}\,}
and
q a ⋅ a = q b ⋅ b = q c ⋅ c = 2 3 T {\displaystyle q_{a}\cdot a=q_{b}\cdot b=q_{c}\cdot c={\frac {2}{3}}T\,}
for area T.
Circumcenter, incenter, and orthocenter
Main articles: Circumcenter, Incenter, and Orthocenter
Carnot's Theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the inradius.[25]:p.83 Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle. This method is especially useful for deducing the properties of more abstract forms of triangles, such as the ones induced by Lie algebras, that otherwise have the same properties as usual triangles.
Euler's theorem states that the distance d between the circumcenter and the incenter is given by[25]:p.85
d 2 = R ( R − 2 r ) {\displaystyle \displaystyle d^{2}=R(R-2r)}
or equivalently
1 R − d + 1 R + d = 1 r , {\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},}
where R is the circumradius and r is the inradius. Thus for all triangles R ≥ 2r, with equality holding for equilateral triangles.
If we denote that the orthocenter divides one altitude into segments of lengths u and v, another altitude into segment lengths w and x, and the third altitude into segment lengths y and z, then uv = wx = yz.[25]:p.94
The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter.[25]:p.99
The sum of the squares of the distances from the vertices to the orthocenter H plus the sum of the squares of the sides equals twelve times the square of the circumradius:[25]:p.102
A H 2 + B H 2 + C H 2 + a 2 + b 2 + c 2 = 12 R 2 . {\displaystyle AH^{2}+BH^{2}+CH^{2}+a^{2}+b^{2}+c^{2}=12R^{2}.}
Angles
In addition to the law of sines, the law of cosines, the law of tangents, and the trigonometric existence conditions given earlier, for any triangle
a = b cos ⁡ C + c cos ⁡ B , b = c cos ⁡ A + a cos ⁡ C , c = a cos ⁡ B + b cos ⁡ A . {\displaystyle a=b\cos C+c\cos B,\quad b=c\cos A+a\cos C,\quad c=a\cos B+b\cos A.}
Morley's trisector theorem
Main article: Morley's trisector theorem
The Morley triangle, resulting from the trisection of each interior angle. This is an example of a finite subdivision rule.
Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the Morley triangle.
Figures inscribed in a triangle
Conics
As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides.
Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse.[30] This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle.
The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles.
For any ellipse inscribed in a triangle ABC, let the foci be P and Q. Then[31]
P A ¯ ⋅ Q A ¯ C A ¯ ⋅ A B ¯ + P B ¯ ⋅ Q B ¯ A B ¯ ⋅ B C ¯ + P C ¯ ⋅ Q C ¯ B C ¯ ⋅ C A ¯ = 1. {\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.}
Convex polygon
Every convex polygon with area T can be inscribed in a triangle of area at most equal to 2T. Equality holds (exclusively) for a parallelogram.[32]
Hexagon
The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point. In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.
Squares
Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has side of length qa and the triangle has a side of length a, part of which side coincides with a side of the square, then qa, a, the altitude ha from the side a, and the triangle's area T are related according to[33][34]
q a = 2 T a a 2 + 2 T = a h a a + h a . {\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.}
The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when a2 = 2T, q = a/2, and the altitude of the triangle from the base of length a is equal to a. The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2 2 / 3 = 0.94.... {\displaystyle 2{\sqrt {2}}/3=0.94....} [34] Both of these extreme cases occur for the isosceles right triangle.
Triangles
From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle.
The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).
Figures circumscribed about a triangle
The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices.
As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides.
Further, every triangle has a unique Steiner circumellipse, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area.
The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its centroid, and its circumcenter.
Of all triangles contained in a given convex polygon, there exists a triangle with maximal area whose vertices are all vertices of the given polygon.[35]
Specifying the location of a point in a triangle
One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in the Cartesian plane, and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane.
Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle:
Trilinear coordinates specify the relative distances of a point from the sides, so that coordinates x : y : z {\displaystyle x:y:z} indicate that the ratio of the distance of the point from the first side to its distance from the second side is x : y {\displaystyle x:y} , etc.
Barycentric coordinates of the form α : β : γ {\displaystyle \alpha :\beta :\gamma } specify the point's location by the relative weights that would have to be put on the three vertices in order to balance the otherwise weightless triangle on the given point.
Non-planar triangles
A non-planar triangle is a triangle which is not contained in a (flat) plane. Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.
While the measures of the internal angles in planar triangles always sum to 180°, a hyperbolic triangle has measures of angles that sum to less than 180°, and a spherical triangle has measures of angles that sum to more than 180°. A hyperbolic triangle can be obtained by drawing on a negatively curved surface, such as a saddle surface, and a spherical triangle can be obtained by drawing on a positively curved surface such as a sphere. Thus, if one draws a giant triangle on the surface of the Earth, one will find that the sum of the measures of its angles is greater than 180°; in fact it will be between 180° and 540°.[36] In particular it is possible to draw a triangle on a sphere such that the measure of each of its internal angles is equal to 90°, adding up to a total of 270°.
Specifically, on a sphere the sum of the angles of a triangle is
180° × (1 + 4f),
where f is the fraction of the sphere's area which is enclosed by the triangle. For example, suppose that we draw a triangle on the Earth's surface with vertices at the North Pole, at a point on the equator at 0° longitude, and a point on the equator at 90° West longitude. The great circle line between the latter two points is the equator, and the great circle line between either of those points and the North Pole is a line of longitude; so there are right angles at the two points on the equator. Moreover, the angle at the North Pole is also 90° because the other two vertices differ by 90° of longitude. So the sum of the angles in this triangle is 90° + 90° + 90° = 270°. The triangle encloses 1/4 of the northern hemisphere (90°/360° as viewed from the North Pole) and therefore 1/8 of the Earth's surface, so in the formula f = 1/8; thus the formula correctly gives the sum of the triangle's angles as 270°.
From the above angle sum formula we can also see that the Earth's surface is locally flat: If we draw an arbitrarily small triangle in the neighborhood of one point on the Earth's surface, the fraction f of the Earth's surface which is enclosed by the triangle will be arbitrarily close to zero. In this case the angle sum formula simplifies to 180°, which we know is what Euclidean geometry tells us for triangles on a flat surface.
Triangles in construction
Main article: Truss
The Flatiron Building in New York is shaped like a triangular prism
Rectangles have been the most popular and common geometric form for buildings since the shape is easy to stack and organize; as a standard, it is easy to design furniture and fixtures to fit inside rectangularly shaped buildings. But triangles, while more difficult to use conceptually, provide a great deal of strength. As computer technology helps architects design creative new buildings, triangular shapes are becoming increasingly prevalent as parts of buildings and as the primary shape for some types of skyscrapers as well as building materials. In Tokyo in 1989, architects had wondered whether it was possible to build a 500-story tower to provide affordable office space for this densely packed city, but with the danger to buildings from earthquakes, architects considered that a triangular shape would have been necessary if such a building was ever to have been built (it hasn't by 2011).[37]
In New York City, as Broadway crisscrosses major avenues, the resulting blocks are cut like triangles, and buildings have been built on these shapes; one such building is the triangularly shaped Flatiron Building which real estate people admit has a "warren of awkward spaces that do not easily accommodate modern office furniture" but that has not prevented the structure from becoming a landmark icon.[38] Designers have made houses in Norway using triangular themes.[39] Triangle shapes have appeared in churches[40] as well as public buildings including colleges[41] as well as supports for innovative home designs.[42]
Triangles are sturdy; while a rectangle can collapse into a parallelogram from pressure to one of its points, triangles have a natural strength which supports structures against lateral pressures. A triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense. Some innovative designers have proposed making bricks not out of rectangles, but with triangular shapes which can be combined in three dimensions.[43] It is likely that triangles will be used increasingly in new ways as architecture increases in complexity. It is important to remember that triangles are strong in terms of rigidity, but while packed in a tessellating arrangement triangles are not as strong as hexagons under compression (hence the prevalence of hexagonal forms in nature). Tessellated triangles still maintain superior strength for cantilevering however, and this is the basis for one of the strongest man made structures, the tetrahedral truss.
See also
Apollonius' theorem
Congruence (geometry)
Desargues' theorem
Dragon's Eye (symbol)
Fermat point
Hadwiger–Finsler inequality
Heronian triangle
Integer triangle
Law of cosines
Law of sines
Law of tangents
Lester's theorem
List of triangle inequalities
List of triangle topics
Ono's inequality
Pedal triangle
Pedoe's inequality
Pythagorean theorem
Special right triangles
Triangle center
Triangular number
Triangulated category
Triangulation (topology)
Notes
Jump up ^ Euclid defines isosceles triangles based on the number of equal sides, i.e. only two equal sides. An alternative approach defines isosceles triangles based on shared properties, i.e. equilateral triangles are a special case of isosceles triangles. wikt:Isosceles triangle
Jump up ^ The n external angles of any n-sided convex polygon add up to 360 degrees.
Jump up ^ Again, in all cases "mirror images" are also similar.
Jump up ^ All pairs of congruent triangles are also similar; but not all pairs of similar triangles are congruent.
References
Jump up ^ Weisstein, Eric W. "Equilateral Triangle". MathWorld.
^ Jump up to: a b Weisstein, Eric W. "Isosceles Triangle". MathWorld.
Jump up ^ Weisstein, Eric W. "Scalene triangle". MathWorld.
Jump up ^ Zeidler, Eberhard (2004). Oxford Users' Guide to Mathematics. Oxford University Press. p. 729. ISBN 978-0-19-850763-5.
Jump up ^ "Euclid's Elements, Book I, Proposition 32".
^ Jump up to: a b Vardan Verdiyan & Daniel Campos Salas, "Simple trigonometric substitutions with broad results", Mathematical Reflections no 6, 2007.
^ Jump up to: a b Longuet-Higgins, Michael S., "On the ratio of the inradius to the circumradius of a triangle", Mathematical Gazette 87, March 2003, 119–120.
^ Jump up to: a b Prof. David E. Joyce. "The Laws of Cosines and Sines". Clark University. Retrieved 1 November 2008.
Jump up ^ Weisstein, Eric W. "Law of Tangents". Wolfram MathWorld. Retrieved 26 July 2012.
Jump up ^ Weisstein, Eric W. "Triangle area". MathWorld.
Jump up ^ Weisstein, Eric W. "Triangle". MathWorld.
Jump up ^ Bart Braden (1986). "The Surveyor's Area Formula" (PDF). The College Mathematics Journal. 17 (4): 326–337. doi:10.2307/2686282.
Jump up ^ Benyi, Arpad, "A Heron-type formula for the triangle," Mathematical Gazette" 87, July 2003, 324–326.
Jump up ^ Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," Mathematical Gazette 89, November 2005, 494.
Jump up ^ Mitchell, Douglas W., "A Heron-type area formula in terms of sines," Mathematical Gazette 93, March 2009, 108–109.
Jump up ^ Sa ́ndor Nagydobai Kiss, "A Distance Property of the Feuerbach Point and Its Extension", Forum Geometricorum 16, 2016, 283–290.
Jump up ^ "Circumradius". AoPSWiki. Archived from the original on 20 June 2013. Retrieved 26 July 2012.
Jump up ^ Mitchell, Douglas W., "The area of a quadrilateral," Mathematical Gazette 93, July 2009, 306–309.
Jump up ^ Pathan, Alex, and Tony Collyer, "Area properties of triangles revisited," Mathematical Gazette 89, November 2005, 495–497.
Jump up ^ Baker, Marcus, "A collection of formulae for the area of a plane triangle," Annals of Mathematics, part 1 in vol. 1(6), January 1885, 134–138; part 2 in vol. 2(1), September 1885, 11–18. The formulas given here are #9, #39a, #39b, #42, and #49. The reader is advised that several of the formulas in this source are not correct.
Jump up ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
Jump up ^ Rosenberg, Steven; Spillane, Michael; and Wulf, Daniel B. "Heron triangles and moduli spaces", Mathematics Teacher 101, May 2008, 656–663.
Jump up ^ Posamentier, Alfred S., and Lehmann, Ingmar, The Secrets of Triangles, Prometheus Books, 2012.
Jump up ^ Dunn, J.A., and Pretty, J.E., "Halving a triangle," Mathematical Gazette 56, May 1972, 105–108.
^ Jump up to: a b c d e f g h Altshiller-Court, Nathan, College Geometry, Dover, 2007.
Jump up ^ Oxman, Victor. "On the existence of triangles with given lengths of one side and two adjacent angle bisectors", Forum Geometricorum 4, 2004, 215–218.
Jump up ^ Mitchell, Douglas W. (2013), "Perpendicular Bisectors of Triangle Sides", Forum Geometricorum 13, 53-59.
^ Jump up to: a b c Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ. Co., 2007
Jump up ^ Altshiller-Court (1925, pp. 70–71)
Jump up ^ Kalman, Dan. "An Elementary Proof of Marden's Theorem", 2008, American Mathematical Monthly 115, 330–338.
Jump up ^ Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161–165.
Jump up ^ Weisstein, Eric W. "Triangle Circumscribing". Wolfram Math World.
Jump up ^ Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles", Mathematics Magazine 71(4), 1998, 278–284.
^ Jump up to: a b Victor Oxman and Moshe Stupel, "Why Are the Side Lengths of the Squares Inscribed in a Triangle so Close to Each Other?", Forum Geometricorum 13 (2013) 113–115.
Jump up ^ -, Christos. "Is the area of intersection of convex polygons always convex?". Math Stack Exchange.
Jump up ^ Watkins, Matthew, Useful Mathematical and Physical Formulae, Walker and Co., 2000.
Jump up ^ Associated Press (10 November 1989). "Tokyo Designers Envision 500-Story Tower". Los Angeles Times. Retrieved 5 March 2011. A construction company said Thursday that it has designed a 500-story skyscraper for Tokyo, ... The building is shaped like a triangle, becoming smaller at the top to help it absorb shock waves. It would have a number of tunnels to let typhoon winds pass through rather than hitting the building with full force.
Jump up ^ Stapinski, Helene (26 May 2010). "A Quirky Building That Has Charmed Its Tenants". The New York Times. Retrieved 5 March 2011. Though it is hard to configure office space in a triangle
Jump up ^ Jodidio, Philip (2009). "Triangle House in Norway". Architecture Week. Retrieved 5 March 2011. Local zoning restrictions determined both the plan and the height of the Triangle House in Nesodden, Norway, which offers views toward the sea through a surrounding pine forest.
Jump up ^ Metz, Tracy (July 2009). "The Chapel of the Deaconesses of Reuilly". Architectural Record. Retrieved 5 March 2011. the classical functions of a church in two pure forms: a stark triangle of glass and, inside it, a rounded, egglike structure made of wood.
Jump up ^ Deborah Snoonian, P.E. (5 March 2011). "Tech Briefs: Seismic framing technology and smart siting aid a California community college". Architectural Record. Retrieved 5 March 2011. More strength, less material ... They share a common material language of structural steel, glass and metal panels, and stucco cladding; their angular, dynamic volumes, folded roof plates, and triangular forms are meant to suggest the plate tectonics of the shifting ground planes they sit on.
Jump up ^ Sarah Amelar (November 2006). "Prairie Ridge Ecostation for Wildlife and Learning". Architectural Record. Retrieved 5 March 2011. Perched like a tree house, the $300,000 structure sits lightly on the terrain, letting the land flow beneath it. Much of the building rests on three triangular heavy-timber frames on a concrete pad.
Jump up ^ Joshua Rothman (13 March 2011). "Building a better brick". Boston Globe. Retrieved 5 March 2011. Bricks are among the world’s oldest building materials — the first were used as long ago as 7,500 B.C. ... An especially beautiful proposal by Rizal Muslimin at the Massachusetts Institute of Technology came in as a runner-up: BeadBricks are flat, triangular bricks that can be combined in three dimensions (rather than the usual two).
External links
Wikimedia Commons has media related to Triangles.
Look up triangle in Wiktionary, the free dictionary.
Ivanov, A.B. (2001) [1994], "Triangle", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Clark Kimberling: Encyclopedia of triangle centers. Lists some 5200 interesting points associated with any triangle.
[hide]
v
t
e
Polygons
Regular
List
1–10 sides
Monogon
Digon
Triangle
Equilateral
Isosceles
Quadrilateral
Square
Rectangle
Rhombus
Parallelogram
Trapezoid
Kite
Pentagon
Hexagon
Heptagon
Octagon
Nonagon (Enneagon)
Decagon
11–20 sides
Hendecagon
Dodecagon
Tridecagon
Tetradecagon
Pentadecagon
Hexadecagon
Heptadecagon
Octadecagon
Enneadecagon
Icosagon
21–100 sides
(selected)
Icosidigon (22)
Icositetragon (24)
Icosihexagon (26)
Icosioctagon (28)
Triacontagon (30)
Triacontadigon (32)
Triacontatetragon (34)
Tetracontagon (40)
Tetracontadigon (42)
Tetracontaoctagon (48)
Pentacontagon (50)
Hexacontagon (60)
Hexacontatetragon (64)
Heptacontagon (70)
Octacontagon (80)
Enneacontagon (90)
Enneacontahexagon (96)
Hectogon (100)
>100 sides
120-gon
257-gon
360-gon
Chiliagon (1,000)
Myriagon (10,000)
65537-gon
Megagon (1,000,000)
Apeirogon (∞)
Star polygons
(5–12 sides)
Pentagram
Hexagram
Heptagram
Octagram
Enneagram
Decagram
Hendecagram
Dodecagram
Retrieved from "https://en.wikipedia.org/w/index.php?title=Triangle&oldid=835178499"
Categories:
Triangles
Elementary shapes
Triangle geometry
Hidden categories:
Wikipedia pages semi-protected against vandalism
Wikipedia indefinitely move-protected pages
Use dmy dates from August 2014
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
View source
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
In other projects
Wikimedia Commons
Wikibooks
Languages
Адыгэбзэ
Afrikaans
አማርኛ
العربية
Aragonés
ܐܪܡܝܐ
অসমীয়া
Asturianu
Aymar aru
Azərbaycanca
تۆرکجه
বাংলা
Bân-lâm-gú
Башҡортса
Беларуская
Беларуская (тарашкевіца)‎
भोजपुरी
Български
Boarisch
Bosanski
Brezhoneg
Català
Чӑвашла
Čeština
ChiShona
Corsu
Cymraeg
Dansk
Davvisámegiella
Deutsch
Dolnoserbski
Eesti
Ελληνικά
Español
Esperanto
Euskara
فارسی
Føroyskt
Français
Gaeilge
Gaelg
Galego
贛語
ગુજરાતી
客家語/Hak-kâ-ngî
한국어
Հայերեն
हिन्दी
Hornjoserbsce
Hrvatski
Ido
Bahasa Indonesia
Íslenska
Italiano
עברית
Basa Jawa
ಕನ್ನಡ
ქართული
Kaszëbsczi
Қазақша
Kiswahili
Kreyòl ayisyen
Kurdî
ລາວ
Latina
Latviešu
Lietuvių
Limburgs
Lingála
Lumbaart
Magyar
Македонски
മലയാളം
मराठी
مصرى
Bahasa Melayu
Mìng-dĕ̤ng-ngṳ̄
Монгол
မြန်မာဘာသာ
Nederlands
नेपाली
नेपाल भाषा
日本語
Norsk
Norsk nynorsk
Nouormand
Occitan
Олык марий
ଓଡ଼ିଆ
Oʻzbekcha/ўзбекча
ਪੰਜਾਬੀ
Pälzisch
پنجابی
پښتو
Patois
ភាសាខ្មែរ
Polski
Português
Română
Runa Simi
Русиньскый
Русский
Scots
Shqip
Sicilianu
Simple English
Slovenčina
Slovenščina
Ślůnski
Soomaaliga
کوردی
Српски / srpski
Srpskohrvatski / српскохрватски
Basa Sunda
Suomi
Svenska
Tagalog
தமிழ்
Татарча/tatarça
తెలుగు
ไทย
Тоҷикӣ
ᏣᎳᎩ
Türkçe
Українська
اردو
Vèneto
Tiếng Việt
文言
West-Vlams
Winaray
吴语
ייִדיש
Yorùbá
粵語
Žemaitėška
中文
128 more
Edit links
This page was last edited on 7 April 2018, at 01:27.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

div
id class
mw-page-base noprint
html
local
div
id class
mw-head-base noprint
html
local
a
id
top
html
local
div
id class
siteNotice centralNotice mw-body-content
html
local
img
alt src width height srcset data-file-width data-file-height
Page semi-protected //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Padlock-silver.svg/20px-Padlock-silver.svg.png 20 20 //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Padlock-silver.svg/30px-Padlock-silver.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Padlock-silver.svg/40px-Padlock-silver.svg.png 2x 128 128
html
local
a
href title
/wiki/Wikipedia:Protection_policy#semi This article is semi-protected due to vandalism
html
local
div
class id
mw-indicators mw-body-content mw-indicator mw-indicator-pp-default
html
local
h1
id class lang
firstHeading firstHeading en
html
local
div
id class
siteSub noprint
html
local
div
id
contentSub
html
local
a
href
#mw-head
html
local
a
href
#p-search
html
local
div
id class
jump-to-nav mw-jump
html
local
a
href class title
/wiki/Triangle_(disambiguation) mw-disambig Triangle (disambiguation)
html
local
div
role class
note hatnote navigation-not-searchable
html
local
th
colspan style
2 text-align:center;font-size:125%;font-weight:bold;background:#e7dcc3;
html
local
tr
html
local
img
alt src width height srcset data-file-width data-file-height
Regular polygon 3 annotated.svg //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Regular_polygon_3_annotated.svg/220px-Regular_polygon_3_annotated.svg.png 220 223 //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Regular_polygon_3_annotated.svg/330px-Regular_polygon_3_annotated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Regular_polygon_3_annotated.svg/440px-Regular_polygon_3_annotated.svg.png 2x 503 509
html
local
a
href class
/wiki/File:Regular_polygon_3_annotated.svg image
html
local
div
html
local
td
colspan style
2 text-align:center
html
local
tr
html
local
th
scope
row
html
local
a
href title
/wiki/Regular_polygon Regular polygon
html
local
td
html
local
tr
html
local
a
href title
/wiki/Edge_(geometry) Edge (geometry)
html
local
a
href title
/wiki/Vertex_(geometry) Vertex (geometry)
html
local
th
scope
row
html
local
td
html
local
tr
html
local
a
href title
/wiki/Schl%C3%A4fli_symbol Schläfli symbol
html
local
th
scope
row
html
local
td
html
local
tr
html
local
a
href class title
/wiki/Coxeter_diagram mw-redirect Coxeter diagram
html
local
th
scope
row
html
local
img
alt src width height data-file-width data-file-height
CDel node 1.png //upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png 9 23 9 23
html
local
img
alt src width height data-file-width data-file-height
CDel 3.png //upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png 6 23 6 23
html
local
img
alt src width height data-file-width data-file-height
CDel node.png //upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png 5 23 5 23
html
local
span
style
display:inline-block;
html
local
td
html
local
tr
html
local
a
href title
/wiki/List_of_planar_symmetry_groups#Rosette_groups List of planar symmetry groups
html
local
th
scope
row
html
local
a
href class title
/wiki/Dihedral_symmetry mw-redirect Dihedral symmetry
html
local
sub
html
local
td
html
local
tr
html
local
a
href class title
/wiki/Internal_angle mw-redirect Internal angle
html
local
a
href title
/wiki/Degree_(angle) Degree (angle)
html
local
th
scope
row
html
local
td
html
local
tr
html
local
a
href title
/wiki/Dual_polygon Dual polygon
html
local
th
scope
row
html
local
td
html
local
tr
html
local
th
scope
row
html
local
a
href title
/wiki/Convex_polygon Convex polygon
html
local
a
href class title
/wiki/Cyclic_polygon mw-redirect Cyclic polygon
html
local
a
href title
/wiki/Equilateral_polygon Equilateral polygon
html
local
a
href title
/wiki/Isogonal_figure Isogonal figure
html
local
a
href title
/wiki/Isotoxal_figure Isotoxal figure
html
local
td
html
local
tr
html
local
tbody
html
local
table
class style
infobox width:22em
html
local
th
colspan style
2 text-align:center;font-size:125%;font-weight:bold;background:#e7dcc3;
html
local
tr
html
local
img
alt src width height srcset data-file-width data-file-height
Triangle illustration.svg //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Triangle_illustration.svg/220px-Triangle_illustration.svg.png 220 244 //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Triangle_illustration.svg/330px-Triangle_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Triangle_illustration.svg/440px-Triangle_illustration.svg.png 2x 450 500
html
local
a
href class
/wiki/File:Triangle_illustration.svg image
html
local
div
html
local
td
colspan style
2 text-align:center
html
local
tr
html
local
a
href title
/wiki/Edge_(geometry) Edge (geometry)
html
local
a
href title
/wiki/Vertex_(geometry) Vertex (geometry)
html
local
th
scope
row
html
local
td
html
local
tr
html
local
a
href title
/wiki/Schl%C3%A4fli_symbol Schläfli symbol
html
local
th
scope
row
html
local
td
html
local
tr
html
local
a
href title
/wiki/Area Area
html
local
th
scope
row
html
local
br
html
local
a
href
#Computing_the_area_of_a_triangle
html
local
td
html
local
tr
html
local
a
href class title
/wiki/Internal_angle mw-redirect Internal angle
html
local
a
href title
/wiki/Degree_(angle) Degree (angle)
html
local
th
scope
row
html
local
td
html
local
tr
html
local
tbody
html
local
table
class style
infobox width:22em
html
local
b
html
local
a
href title
/wiki/Polygon Polygon
html
local
a
href title
/wiki/Edge_(geometry) Edge (geometry)
html
local
a
href title
/wiki/Vertex_(geometry) Vertex (geometry)
html
local
a
href title
/wiki/Shape Shape
html
local
a
href title
/wiki/Geometry Geometry
html
local
i
html
local
i
html
local
i
html
local
mi
mathvariant
normal
html
local
mi
html
local
mi
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \triangle ABC}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/821677f03b63c3c2e448dffc2ae9c8eea31d9d48 mwe-math-fallback-image-inline true vertical-align: -0.338ex; width:7.339ex; height:2.176ex; \triangle ABC
html
local
span
class
mwe-math-element
html
local
p
html
local
a
href title
/wiki/Euclidean_geometry Euclidean geometry
html
local
a
href class title
https://en.wiktionary.org/wiki/collinear extiw wikt:collinear
html
local
a
href class title
/wiki/Plane_(mathematics) mw-redirect Plane (mathematics)
html
local
a
href title
/wiki/Euclidean_space Euclidean space
html
local
a
href class title
/wiki/Euclidean_plane mw-redirect Euclidean plane
html
local
p
html
local
p
html
local
h2
html
local
a
role tabindex class
button 0 togglelink
html
local
span
class
toctoggle
html
local
div
class dir xml:lang
toctitle ltr en
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Types_of_triangle
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#By_lengths_of_sides
html
local
li
class
toclevel-2 tocsection-2
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#By_internal_angles
html
local
li
class
toclevel-2 tocsection-3
html
local
ul
html
local
li
class
toclevel-1 tocsection-1
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Basic_facts
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Similarity_and_congruence
html
local
li
class
toclevel-2 tocsection-5
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Right_triangles
html
local
li
class
toclevel-2 tocsection-6
html
local
ul
html
local
li
class
toclevel-1 tocsection-4
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Existence_of_a_triangle
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Condition_on_the_sides
html
local
li
class
toclevel-2 tocsection-8
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Conditions_on_the_angles
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Trigonometric_conditions
html
local
li
class
toclevel-3 tocsection-10
html
local
ul
html
local
li
class
toclevel-2 tocsection-9
html
local
ul
html
local
li
class
toclevel-1 tocsection-7
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Points,_lines,_and_circles_associated_with_a_triangle
html
local
li
class
toclevel-1 tocsection-11
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Computing_the_sides_and_angles
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Trigonometric_ratios_in_right_triangles
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Sine,_cosine_and_tangent
html
local
li
class
toclevel-3 tocsection-14
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Inverse_functions
html
local
li
class
toclevel-3 tocsection-15
html
local
ul
html
local
li
class
toclevel-2 tocsection-13
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Sine,_cosine_and_tangent_rules
html
local
li
class
toclevel-2 tocsection-16
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Solution_of_triangles
html
local
li
class
toclevel-2 tocsection-17
html
local
ul
html
local
li
class
toclevel-1 tocsection-12
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Computing_the_area_of_a_triangle
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Using_trigonometry
html
local
li
class
toclevel-2 tocsection-19
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Using_Heron's_formula
html
local
li
class
toclevel-2 tocsection-20
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Using_vectors
html
local
li
class
toclevel-2 tocsection-21
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Using_coordinates
html
local
li
class
toclevel-2 tocsection-22
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Using_line_integrals
html
local
li
class
toclevel-2 tocsection-23
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Formulas_resembling_Heron's_formula
html
local
li
class
toclevel-2 tocsection-24
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Using_Pick's_theorem
html
local
li
class
toclevel-2 tocsection-25
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Other_area_formulas
html
local
li
class
toclevel-2 tocsection-26
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Upper_bound_on_the_area
html
local
li
class
toclevel-2 tocsection-27
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Bisecting_the_area
html
local
li
class
toclevel-2 tocsection-28
html
local
ul
html
local
li
class
toclevel-1 tocsection-18
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Further_formulas_for_general_Euclidean_triangles
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Medians,_angle_bisectors,_perpendicular_side_bisectors,_and_altitudes
html
local
li
class
toclevel-2 tocsection-30
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Circumradius_and_inradius
html
local
li
class
toclevel-2 tocsection-31
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Adjacent_triangles
html
local
li
class
toclevel-2 tocsection-32
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Centroid
html
local
li
class
toclevel-2 tocsection-33
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Circumcenter,_incenter,_and_orthocenter
html
local
li
class
toclevel-2 tocsection-34
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Angles
html
local
li
class
toclevel-2 tocsection-35
html
local
ul
html
local
li
class
toclevel-1 tocsection-29
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Morley's_trisector_theorem
html
local
li
class
toclevel-1 tocsection-36
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Figures_inscribed_in_a_triangle
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Conics
html
local
li
class
toclevel-2 tocsection-38
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Convex_polygon
html
local
li
class
toclevel-2 tocsection-39
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Hexagon
html
local
li
class
toclevel-2 tocsection-40
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Squares
html
local
li
class
toclevel-2 tocsection-41
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Triangles
html
local
li
class
toclevel-2 tocsection-42
html
local
ul
html
local
li
class
toclevel-1 tocsection-37
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Figures_circumscribed_about_a_triangle
html
local
li
class
toclevel-1 tocsection-43
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Specifying_the_location_of_a_point_in_a_triangle
html
local
li
class
toclevel-1 tocsection-44
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Non-planar_triangles
html
local
li
class
toclevel-1 tocsection-45
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Triangles_in_construction
html
local
li
class
toclevel-1 tocsection-46
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#See_also
html
local
li
class
toclevel-1 tocsection-47
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#Notes
html
local
li
class
toclevel-1 tocsection-48
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#References
html
local
li
class
toclevel-1 tocsection-49
html
local
span
class
tocnumber
html
local
span
class
toctext
html
local
a
href
#External_links
html
local
li
class
toclevel-1 tocsection-50
html
local
ul
html
local
div
id class
toc toc
html
local
p
html
local
span
class id
mw-headline Types_of_triangle
html
local
h2
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euler_diagram_of_triangle_types.svg/320px-Euler_diagram_of_triangle_types.svg.png 320 128 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euler_diagram_of_triangle_types.svg/480px-Euler_diagram_of_triangle_types.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Euler_diagram_of_triangle_types.svg/640px-Euler_diagram_of_triangle_types.svg.png 2x 512 205
html
local
a
href class
/wiki/File:Euler_diagram_of_triangle_types.svg image
html
local
a
href class title
/wiki/File:Euler_diagram_of_triangle_types.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href title
/wiki/Euler_diagram Euler diagram
html
local
i
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:322px;
html
local
span
class id
mw-headline By_lengths_of_sides
html
local
h3
html
local
p
html
local
a
href title
/wiki/Equilateral_triangle Equilateral triangle
html
local
i
html
local
a
href title
/wiki/Regular_polygon Regular polygon
html
local
a
href
#cite_note-1
html
local
sup
id class
cite_ref-1 reference
html
local
li
html
local
a
href title
/wiki/Isosceles_triangle Isosceles triangle
html
local
i
html
local
a
href
#cite_note-2
html
local
sup
id class
cite_ref-2 reference
html
local
a
href
#cite_note-MWisosceles-3
html
local
sup
id class
cite_ref-MWisosceles_3-0 reference
html
local
a
href class title
/wiki/Isosceles_triangle_theorem mw-redirect Isosceles triangle theorem
html
local
a
href title
/wiki/Euclid Euclid
html
local
i
html
local
a
href
#cite_note-MWisosceles-3
html
local
sup
id class
cite_ref-MWisosceles_3-1 reference
html
local
a
href title
/wiki/Tetrakis_square_tiling Tetrakis square tiling
html
local
li
html
local
i
html
local
a
href
#cite_note-4
html
local
sup
id class
cite_ref-4 reference
html
local
li
html
local
ul
html
local
img
alt src width height srcset data-file-width data-file-height
Equilateral Triangle //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/122px-Triangle.Equilateral.svg.png 122 110 //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/183px-Triangle.Equilateral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Triangle.Equilateral.svg/244px-Triangle.Equilateral.svg.png 2x 122 110
html
local
a
href class title
/wiki/File:Triangle.Equilateral.svg image Equilateral Triangle
html
local
td
html
local
img
alt src width height srcset data-file-width data-file-height
Isosceles triangle //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/74px-Triangle.Isosceles.svg.png 74 114 //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/111px-Triangle.Isosceles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Triangle.Isosceles.svg/148px-Triangle.Isosceles.svg.png 2x 74 114
html
local
a
href class title
/wiki/File:Triangle.Isosceles.svg image Isosceles triangle
html
local
td
width
125
html
local
img
alt src width height srcset data-file-width data-file-height
Scalene triangle //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/245px-Triangle.Scalene.svg.png 245 110 //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/368px-Triangle.Scalene.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Scalene.svg/490px-Triangle.Scalene.svg.png 2x 245 110
html
local
a
href class title
/wiki/File:Triangle.Scalene.svg image Scalene triangle
html
local
td
html
local
tr
html
local
td
html
local
td
html
local
td
html
local
tr
html
local
tbody
html
local
table
style
margin:auto; text-align:center
html
local
a
href title
/wiki/Hatch_mark#Congruency_notation Hatch mark
html
local
a
href title
/wiki/Tally_marks Tally marks
html
local
p
html
local
span
class id
mw-headline By_internal_angles
html
local
h3
html
local
a
href class title
/wiki/Internal_angle mw-redirect Internal angle
html
local
a
href title
/wiki/Degree_(angle) Degree (angle)
html
local
p
html
local
a
href title
/wiki/Right_triangle Right triangle
html
local
i
html
local
i
html
local
i
html
local
a
href title
/wiki/Angle Angle
html
local
a
href title
/wiki/Hypotenuse Hypotenuse
html
local
i
html
local
i
html
local
a
href
#cite_note-5
html
local
sup
id class
cite_ref-5 reference
html
local
a
href class title
https://en.wiktionary.org/wiki/cathetus extiw wikt:cathetus
html
local
i
html
local
a
href title
/wiki/Pythagorean_theorem Pythagorean theorem
html
local
i
html
local
sup
html
local
i
html
local
sup
html
local
i
html
local
sup
html
local
span
class
nowrap
html
local
i
html
local
i
html
local
i
html
local
a
href class title
/wiki/Special_right_triangles mw-redirect Special right triangles
html
local
sup
html
local
sup
html
local
sup
html
local
span
class
nowrap
html
local
a
href title
/wiki/Pythagorean_triple Pythagorean triple
html
local
li
html
local
a
href class title
/wiki/Oblique_triangle mw-redirect Oblique triangle
html
local
li
html
local
a
href class title
/wiki/Acute_triangle mw-redirect Acute triangle
html
local
i
html
local
i
html
local
i
html
local
sup
html
local
i
html
local
sup
html
local
i
html
local
sup
html
local
span
class
nowrap
html
local
i
html
local
i
html
local
li
html
local
a
href class title
/wiki/Obtuse_triangle mw-redirect Obtuse triangle
html
local
i
html
local
i
html
local
i
html
local
sup
html
local
i
html
local
sup
html
local
i
html
local
sup
html
local
span
class
nowrap
html
local
i
html
local
i
html
local
li
html
local
a
href class title
https://en.wiktionary.org/wiki/collinear extiw wikt:collinear
html
local
a
href title
/wiki/Degeneracy_(mathematics) Degeneracy (mathematics)
html
local
li
html
local
li
html
local
ul
html
local
p
html
local
img
alt src width height srcset data-file-width data-file-height
Right triangle //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/150px-Triangle.Right.svg.png 150 113 //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/225px-Triangle.Right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Triangle.Right.svg/300px-Triangle.Right.svg.png 2x 150 113
html
local
a
href class title
/wiki/File:Triangle.Right.svg image Right triangle
html
local
td
html
local
img
alt src width height srcset data-file-width data-file-height
Obtuse triangle //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/113px-Triangle.Obtuse.svg.png 113 113 //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/170px-Triangle.Obtuse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Triangle.Obtuse.svg/226px-Triangle.Obtuse.svg.png 2x 113 113
html
local
a
href class title
/wiki/File:Triangle.Obtuse.svg image Obtuse triangle
html
local
td
width
185
html
local
img
alt src width height srcset data-file-width data-file-height
Acute triangle //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/181px-Triangle.Acute.svg.png 181 113 //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/272px-Triangle.Acute.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Triangle.Acute.svg/362px-Triangle.Acute.svg.png 2x 181 113
html
local
a
href class title
/wiki/File:Triangle.Acute.svg image Acute triangle
html
local
td
width
185
html
local
tr
html
local
td
html
local
td
html
local
td
html
local
tr
html
local
td
html
local
mspace
width
2em
html
local
mspace
width
2em
html
local
mspace
width
2em
html
local
mspace
width
2em
html
local
mspace
width
2em
html
local
mspace
width
2em
html
local
mrow
html
local
mo
html
local
munder
html
local
mrow
class
MJX-TeXAtom-OP MJX-fixedlimits
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
munder
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \underbrace {\qquad \qquad \qquad \qquad \qquad \qquad } _{}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a39562ec2dc0af250592ab5336422848a084459c mwe-math-fallback-image-inline true vertical-align: -3.505ex; margin-right: -0.028ex; width:27.899ex; height:3.676ex; \underbrace {\qquad \qquad \qquad \qquad \qquad \qquad } _{}
html
local
span
class
mwe-math-element
html
local
td
colspan
2
html
local
tr
html
local
td
html
local
td
colspan
2
html
local
tr
html
local
tbody
html
local
table
style
margin:auto; text-align:center
html
local
span
class id
mw-headline Basic_facts
html
local
h2
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Remint3.svg/300px-Remint3.svg.png 300 149 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Remint3.svg/450px-Remint3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Remint3.svg/600px-Remint3.svg.png 2x 975 483
html
local
a
href class
/wiki/File:Remint3.svg image
html
local
a
href class title
/wiki/File:Remint3.svg internal Enlarge
html
local
div
class
thumbcaption magnify
html
local
div
class style
thumb tright thumbinner width:302px;
html
local
a
href title
/wiki/Dimension Dimension
html
local
a
href class title
/wiki/Plane_figure mw-redirect Plane figure
html
local
a
href
#Non-planar_triangles
html
local
a
href title
/wiki/Simplex Simplex
html
local
i
html
local
a
href title
/wiki/Polytope Polytope
html
local
a
href title
/wiki/Euclid Euclid
html
local
a
href title
/wiki/Euclid%27s_Elements Euclid's Elements
html
local
i
html
local
p
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Triangle_sommeangles.svg/300px-Triangle_sommeangles.svg.png 300 153 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Triangle_sommeangles.svg/450px-Triangle_sommeangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Triangle_sommeangles.svg/600px-Triangle_sommeangles.svg.png 2x 500 255
html
local
a
href class
/wiki/File:Triangle_sommeangles.svg image
html
local
a
href class title
/wiki/File:Triangle_sommeangles.svg internal Enlarge
html
local
div
class
thumbcaption magnify
html
local
div
class style
thumb tright thumbinner width:302px;
html
local
a
href title
/wiki/Sum_of_angles_of_a_triangle Sum of angles of a triangle
html
local
a
href title
/wiki/Euclidean_space Euclidean space
html
local
a
href
#cite_note-6
html
local
sup
id class
cite_ref-6 reference
html
local
a
href title
/wiki/Parallel_postulate Parallel postulate
html
local
a
href class title
/wiki/Exterior_angle mw-redirect Exterior angle
html
local
i
html
local
a
href class title
/wiki/Supplementary_angle mw-redirect Supplementary angle
html
local
a
href title
/wiki/Exterior_angle_theorem Exterior angle theorem
html
local
a
href
#cite_note-7
html
local
sup
id class
cite_ref-7 reference
html
local
p
html
local
span
class id
mw-headline Similarity_and_congruence
html
local
h3
html
local
a
href title
/wiki/Similarity_(geometry) Similarity (geometry)
html
local
i
html
local
p
html
local
a
href title
/wiki/Theorem Theorem
html
local
p
html
local
a
href title
/wiki/If_and_only_if If and only if
html
local
li
html
local
i
html
local
li
html
local
a
href
#cite_note-8
html
local
sup
id class
cite_ref-8 reference
html
local
li
html
local
ul
html
local
a
href title
/wiki/Congruence_(geometry) Congruence (geometry)
html
local
a
href
#cite_note-9
html
local
sup
id class
cite_ref-9 reference
html
local
p
html
local
a
href class title
/wiki/Necessary_and_sufficient_condition mw-redirect Necessary and sufficient condition
html
local
p
html
local
li
html
local
i
html
local
li
html
local
li
html
local
i
html
local
li
html
local
ul
html
local
p
html
local
li
html
local
li
html
local
ul
html
local
p
html
local
i
html
local
li
html
local
ul
html
local
a
href class title
/wiki/Trigonometric_function mw-redirect Trigonometric function
html
local
a
href title
/wiki/Angle Angle
html
local
a
href title
/wiki/Trigonometry Trigonometry
html
local
p
html
local
span
class id
mw-headline Right_triangles
html
local
h3
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/220px-Pythagorean.svg.png 220 178 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/330px-Pythagorean.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Pythagorean.svg/440px-Pythagorean.svg.png 2x 265 214
html
local
a
href class
/wiki/File:Pythagorean.svg image
html
local
a
href class title
/wiki/File:Pythagorean.svg internal Enlarge
html
local
div
class
thumbcaption magnify
html
local
div
class style
thumb tright thumbinner width:222px;
html
local
a
href title
/wiki/Pythagorean_theorem Pythagorean theorem
html
local
a
href title
/wiki/Right_triangle Right triangle
html
local
a
href title
/wiki/Hypotenuse Hypotenuse
html
local
i
html
local
i
html
local
i
html
local
p
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle a^{2}+b^{2}=c^{2}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/90b56b985c78deb115014efe90ce634d73dd51fa mwe-math-fallback-image-inline true vertical-align: -0.505ex; width:12.983ex; height:2.843ex; {\displaystyle a^{2}+b^{2}=c^{2}.}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
p
html
local
p
html
local
a
href class title
/wiki/Complementary_angles mw-redirect Complementary angles
html
local
li
html
local
ul
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mo
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mo
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mo
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mn
html
local
mo
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle a+b+90^{\circ }=180^{\circ }\Rightarrow a+b=90^{\circ }\Rightarrow a=90^{\circ }-b.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4a4e4937341a3ec439dc2709074359502c6ab3 mwe-math-fallback-image-inline true vertical-align: -0.505ex; width:49.893ex; height:2.509ex; {\displaystyle a+b+90^{\circ }=180^{\circ }\Rightarrow a+b=90^{\circ }\Rightarrow a=90^{\circ }-b.}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
style
border-top:1px solid; padding:0 0.1em;
html
local
span
class
nowrap
html
local
li
html
local
span
style
border-top:1px solid; padding:0 0.1em;
html
local
span
class
nowrap
html
local
li
html
local
ul
html
local
mi
html
local
mo
html
local
mn
html
local
mi
html
local
mspace
width
thinmathspace
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle c=2a\,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/54e01c38c0bd77eaa0e538ba787554851d1fd92a mwe-math-fallback-image-inline true vertical-align: -0.338ex; width:6.885ex; height:2.176ex; c=2a\,
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle b=a\times {\sqrt {3}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdbf65efc09b148bf9479b3530421825cb11ca4 mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:11.911ex; height:2.843ex; b=a\times {\sqrt {3}}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
dd
html
local
dl
html
local
a
href title
/wiki/Law_of_cosines Law of cosines
html
local
a
href title
/wiki/Law_of_sines Law of sines
html
local
i
html
local
i
html
local
p
html
local
span
class id
mw-headline Existence_of_a_triangle
html
local
h2
html
local
span
class id
mw-headline Condition_on_the_sides
html
local
h3
html
local
a
href title
/wiki/Triangle_inequality Triangle inequality
html
local
p
html
local
span
class id
mw-headline Conditions_on_the_angles
html
local
h3
html
local
p
html
local
span
class id
mw-headline Trigonometric_conditions
html
local
h4
html
local
i
html
local
i
html
local
i
html
local
a
href title
/wiki/If_and_only_if If and only if
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}+\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}=1,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/36c498282e167c138ed480f54fbc78cf867c95d3 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:47.411ex; height:5.343ex; \tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}+\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}=1,
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-VV-10
html
local
sup
id class
cite_ref-VV_10-0 reference
html
local
dd
html
local
dl
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}+2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}=1,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/4b92b13d79afa14b09f7c09e2fd4249c7520952a mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:51.552ex; height:5.343ex; \sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}+2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}=1,
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-VV-10
html
local
sup
id class
cite_ref-VV_10-1 reference
html
local
dd
html
local
dl
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mn
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mn
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mn
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mn
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )=4\sin(\alpha )\sin(\beta )\sin(\gamma ),}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f5d940cbf33f05b2694fe32a6366056a944c79 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:51.391ex; height:2.843ex; \sin(2\alpha )+\sin(2\beta )+\sin(2\gamma )=4\sin(\alpha )\sin(\beta )\sin(\gamma ),
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mn
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha )\cos(\beta )\cos(\gamma )=1,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a61f0d88837e778ed7d717f710f49b006dbe2e14 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:52.335ex; height:3.176ex; \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha )\cos(\beta )\cos(\gamma )=1,
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-LH-11
html
local
sup
id class
cite_ref-LH_11-0 reference
html
local
dd
html
local
dl
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \tan(\alpha )+\tan(\beta )+\tan(\gamma )=\tan(\alpha )\tan(\beta )\tan(\gamma ),}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/4c31be0c0cca5c5b60cf1737d428807225b90780 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:49.378ex; height:2.843ex; \tan(\alpha )+\tan(\beta )+\tan(\gamma )=\tan(\alpha )\tan(\beta )\tan(\gamma ),
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
span
id
Points.2C_lines.2C_and_circles_associated_with_a_triangle
html
local
span
class id
mw-headline Points,_lines,_and_circles_associated_with_a_triangle
html
local
h2
html
local
a
href title
/wiki/Encyclopedia_of_Triangle_Centers Encyclopedia of Triangle Centers
html
local
a
href title
/wiki/Ceva%27s_theorem Ceva's theorem
html
local
a
href title
/wiki/Concurrent_lines Concurrent lines
html
local
a
href class title
https://en.wiktionary.org/wiki/collinear extiw wikt:collinear
html
local
a
href class title
/wiki/Menelaus%27_theorem mw-redirect Menelaus' theorem
html
local
p
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Triangle.Circumcenter.svg/220px-Triangle.Circumcenter.svg.png 220 220 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Triangle.Circumcenter.svg/330px-Triangle.Circumcenter.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Triangle.Circumcenter.svg/440px-Triangle.Circumcenter.svg.png 2x 198 198
html
local
a
href class
/wiki/File:Triangle.Circumcenter.svg image
html
local
a
href class title
/wiki/File:Triangle.Circumcenter.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href class title
/wiki/Circumcenter mw-redirect Circumcenter
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:222px;
html
local
a
href title
/wiki/Bisection Bisection
html
local
a
href title
/wiki/Midpoint Midpoint
html
local
a
href class title
/wiki/Circumcenter mw-redirect Circumcenter
html
local
b
html
local
a
href class title
/wiki/Circumcircle mw-redirect Circumcircle
html
local
a
href title
/wiki/Circle Circle
html
local
i
html
local
i
html
local
p
html
local
a
href class title
/wiki/Thales%27_theorem mw-redirect Thales' theorem
html
local
p
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Orthocenter.svg/220px-Triangle.Orthocenter.svg.png 220 176 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Orthocenter.svg/330px-Triangle.Orthocenter.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Orthocenter.svg/440px-Triangle.Orthocenter.svg.png 2x 182 146
html
local
a
href class
/wiki/File:Triangle.Orthocenter.svg image
html
local
a
href class title
/wiki/File:Triangle.Orthocenter.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href class title
/wiki/Orthocenter mw-redirect Orthocenter
html
local
div
class
thumbcaption
html
local
div
class style
thumb tleft thumbinner width:222px;
html
local
a
href title
/wiki/Altitude_(triangle) Altitude (triangle)
html
local
i
html
local
i
html
local
a
href class title
/wiki/Orthocenter mw-redirect Orthocenter
html
local
b
html
local
p
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle.Incircle.svg/220px-Triangle.Incircle.svg.png 220 190 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle.Incircle.svg/330px-Triangle.Incircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle.Incircle.svg/440px-Triangle.Incircle.svg.png 2x 182 157
html
local
a
href class
/wiki/File:Triangle.Incircle.svg image
html
local
a
href class title
/wiki/File:Triangle.Incircle.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href class title
/wiki/Incircle mw-redirect Incircle
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:222px;
html
local
a
href class title
/wiki/Angle_bisector mw-redirect Angle bisector
html
local
a
href title
/wiki/Incenter Incenter
html
local
b
html
local
a
href class title
/wiki/Incircle mw-redirect Incircle
html
local
i
html
local
a
href class title
/wiki/Excircle mw-redirect Excircle
html
local
a
href title
/wiki/Orthocentric_system Orthocentric system
html
local
p
html
local
div
style
clear:left;
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Triangle.Centroid.svg/220px-Triangle.Centroid.svg.png 220 178 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Triangle.Centroid.svg/330px-Triangle.Centroid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Triangle.Centroid.svg/440px-Triangle.Centroid.svg.png 2x 182 147
html
local
a
href class
/wiki/File:Triangle.Centroid.svg image
html
local
a
href class title
/wiki/File:Triangle.Centroid.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href title
/wiki/Centroid Centroid
html
local
div
class
thumbcaption
html
local
div
class style
thumb tleft thumbinner width:222px;
html
local
a
href title
/wiki/Median_(geometry) Median (geometry)
html
local
a
href title
/wiki/Vertex_(geometry) Vertex (geometry)
html
local
a
href title
/wiki/Midpoint Midpoint
html
local
a
href title
/wiki/Centroid Centroid
html
local
b
html
local
a
href title
/wiki/Center_of_mass Center of mass
html
local
p
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Triangle.NinePointCircle.svg/220px-Triangle.NinePointCircle.svg.png 220 178 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Triangle.NinePointCircle.svg/330px-Triangle.NinePointCircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Triangle.NinePointCircle.svg/440px-Triangle.NinePointCircle.svg.png 2x 182 147
html
local
a
href class
/wiki/File:Triangle.NinePointCircle.svg image
html
local
a
href class title
/wiki/File:Triangle.NinePointCircle.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href title
/wiki/Nine-point_circle Nine-point circle
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:222px;
html
local
a
href title
/wiki/Nine-point_circle Nine-point circle
html
local
a
href class title
/wiki/Orthocenter mw-redirect Orthocenter
html
local
a
href title
/wiki/Nine-point_circle Nine-point circle
html
local
a
href class title
/wiki/Excircle mw-redirect Excircle
html
local
p
html
local
div
style
clear:left;
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle.EulerLine.svg/220px-Triangle.EulerLine.svg.png 220 178 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle.EulerLine.svg/330px-Triangle.EulerLine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Triangle.EulerLine.svg/440px-Triangle.EulerLine.svg.png 2x 520 420
html
local
a
href class
/wiki/File:Triangle.EulerLine.svg image
html
local
a
href class title
/wiki/File:Triangle.EulerLine.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href class title
/wiki/Euler%27s_line mw-redirect Euler's line
html
local
div
class
thumbcaption
html
local
div
class style
thumb tleft thumbinner width:222px;
html
local
a
href class title
/wiki/Euler%27s_line mw-redirect Euler's line
html
local
p
html
local
p
html
local
a
href title
/wiki/Symmedian Symmedian
html
local
a
href class title
/wiki/Symmedian_point mw-redirect Symmedian point
html
local
p
html
local
div
style
clear:both;
html
local
span
class id
mw-headline Computing_the_sides_and_angles
html
local
h2
html
local
p
html
local
span
class id
mw-headline Trigonometric_ratios_in_right_triangles
html
local
h3
html
local
a
href title
/wiki/Trigonometric_functions Trigonometric functions
html
local
div
role class
note hatnote navigation-not-searchable
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/220px-Trigonometry_triangle.svg.png 220 220 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/330px-Trigonometry_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Trigonometry_triangle.svg/440px-Trigonometry_triangle.svg.png 2x 1000 1000
html
local
a
href class
/wiki/File:Trigonometry_triangle.svg image
html
local
a
href class title
/wiki/File:Trigonometry_triangle.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href title
/wiki/Right_triangle Right triangle
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:222px;
html
local
a
href title
/wiki/Right_triangle Right triangle
html
local
p
html
local
a
href title
/wiki/Hypotenuse Hypotenuse
html
local
i
html
local
i
html
local
li
html
local
i
html
local
i
html
local
li
html
local
i
html
local
i
html
local
li
html
local
ul
html
local
span
id
Sine.2C_cosine_and_tangent
html
local
span
class id
mw-headline Sine,_cosine_and_tangent
html
local
h4
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mtext
html
local
mtext
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mspace
width
thinmathspace
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \sin A={\frac {\text{opposite side}}{\text{hypotenuse}}}={\frac {a}{h}}\,.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/6938562ac86376a1c6b4fef739a0e9233db780b9 mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:28.107ex; height:5.843ex; \sin A={\frac {\text{opposite side}}{\text{hypotenuse}}}={\frac {a}{h}}\,.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
a
href class title
/wiki/Similar_triangles mw-redirect Similar triangles
html
local
p
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mtext
html
local
mtext
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mspace
width
thinmathspace
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \cos A={\frac {\text{adjacent side}}{\text{hypotenuse}}}={\frac {b}{h}}\,.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dd6d3eac611567d60ab3cb294e5d9fa99e2f5 mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:28.543ex; height:5.843ex; \cos A={\frac {\text{adjacent side}}{\text{hypotenuse}}}={\frac {b}{h}}\,.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mtext
html
local
mtext
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mspace
width
thinmathspace
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \tan A={\frac {\text{opposite  side}}{\text{adjacent side}}}={\frac {a}{b}}={\frac {\sin A}{\cos A}}\,.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/9269101715cdf1fd07cc5a1cf107ec34232c2755 mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:38.258ex; height:5.843ex; \tan A={\frac {\text{opposite  side}}{\text{adjacent side}}}={\frac {a}{b}}={\frac {\sin A}{\cos A}}\,.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
a
href title
/wiki/Trigonometry#Mnemonics Trigonometry
html
local
a
href title
/wiki/Mnemonic Mnemonic
html
local
p
html
local
span
class id
mw-headline Inverse_functions
html
local
h4
html
local
a
href title
/wiki/Inverse_trigonometric_functions Inverse trigonometric functions
html
local
p
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mo
html
local
mtext
html
local
mtext
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mrow
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \theta =\arcsin \left({\frac {\text{opposite side}}{\text{hypotenuse}}}\right)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/2660fbb5398dcd0928df1ee50ab60e23dd28b430 mwe-math-fallback-image-inline true vertical-align: -2.505ex; width:27.287ex; height:6.176ex; \theta =\arcsin \left({\frac {\text{opposite side}}{\text{hypotenuse}}}\right)
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mo
html
local
mtext
html
local
mtext
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mrow
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \theta =\arccos \left({\frac {\text{adjacent side}}{\text{hypotenuse}}}\right)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/9c857fad1ae590bc9e40f4c1897a453bf341c64a mwe-math-fallback-image-inline true vertical-align: -2.505ex; width:27.724ex; height:6.176ex; \theta =\arccos \left({\frac {\text{adjacent side}}{\text{hypotenuse}}}\right)
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mo
html
local
mtext
html
local
mtext
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mrow
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \theta =\arctan \left({\frac {\text{opposite side}}{\text{adjacent side}}}\right)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e2460e4cf594691bc848b56e5d1d6cc6b9878f mwe-math-fallback-image-inline true vertical-align: -2.505ex; width:27.972ex; height:6.176ex; \theta =\arctan \left({\frac {\text{opposite side}}{\text{adjacent side}}}\right)
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
sup
html
local
sup
html
local
a
href title
/wiki/Multiplicative_inverse Multiplicative inverse
html
local
a
href class title
/wiki/Compositional_inverse mw-redirect Compositional inverse
html
local
p
html
local
span
id
Sine.2C_cosine_and_tangent_rules
html
local
span
class id
mw-headline Sine,_cosine_and_tangent_rules
html
local
h3
html
local
a
href title
/wiki/Law_of_sines Law of sines
html
local
a
href title
/wiki/Law_of_cosines Law of cosines
html
local
a
href title
/wiki/Law_of_tangents Law of tangents
html
local
div
role class
note hatnote navigation-not-searchable
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Triangle_with_notations_2.svg/220px-Triangle_with_notations_2.svg.png 220 132 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Triangle_with_notations_2.svg/330px-Triangle_with_notations_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Triangle_with_notations_2.svg/440px-Triangle_with_notations_2.svg.png 2x 200 120
html
local
a
href class
/wiki/File:Triangle_with_notations_2.svg image
html
local
a
href class title
/wiki/File:Triangle_with_notations_2.svg internal Enlarge
html
local
div
class
thumbcaption magnify
html
local
div
class style
thumb tleft thumbinner width:222px;
html
local
a
href title
/wiki/Law_of_sines Law of sines
html
local
a
href
#cite_note-LawCosSin-12
html
local
sup
id class
cite_ref-LawCosSin_12-0 reference
html
local
p
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/0f94b51f7a15bbf213c3dafdff05fc9c2d246926 mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:23.162ex; height:5.843ex; {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
a
href title
/wiki/Law_of_cosines Law of cosines
html
local
a
href
#cite_note-LawCosSin-12
html
local
sup
id class
cite_ref-LawCosSin_12-1 reference
html
local
p
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
p
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mtext
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/4993a60f1c5a232ac3c0861760a6002a6b788241 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:25.716ex; height:3.176ex; c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mtext
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/75b23b1e839784f2d9571a880655efd3008b89ee mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:25.795ex; height:3.176ex; b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mtext
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
stretchy
false
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/82cc90ccf2a2e7873b8e31c1ef78bbcbdbc2ad8f mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:25.719ex; height:3.176ex; a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mrow
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \alpha =\arccos \left({\frac {b^{2}+c^{2}-a^{2}}{2bc}}\right)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2ac5a61af7051f9985fe9e4fc779ce5e309ce0 mwe-math-fallback-image-inline true vertical-align: -2.505ex; width:27.139ex; height:6.343ex; \alpha =\arccos \left({\frac {b^{2}+c^{2}-a^{2}}{2bc}}\right)
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mrow
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \beta =\arccos \left({\frac {a^{2}+c^{2}-b^{2}}{2ac}}\right)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2679afd1239ce5c8a5cf63a81655f1f15bf0b5 mwe-math-fallback-image-inline true vertical-align: -2.505ex; width:26.983ex; height:6.343ex; \beta =\arccos \left({\frac {a^{2}+c^{2}-b^{2}}{2ac}}\right)
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mrow
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \gamma =\arccos \left({\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/f694b8cdcf769d3fc5f32c2c52fee55cef5aabae mwe-math-fallback-image-inline true vertical-align: -2.505ex; width:26.913ex; height:6.343ex; \gamma =\arccos \left({\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
a
href title
/wiki/Law_of_tangents Law of tangents
html
local
a
href
#cite_note-13
html
local
sup
id class
cite_ref-13 reference
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mrow
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7a4a3592aa66976f1d34bbbb403daa392f9fdf mwe-math-fallback-image-inline true vertical-align: -3.338ex; width:24.266ex; height:7.843ex; {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Solution_of_triangles
html
local
h3
html
local
a
href title
/wiki/Solution_of_triangles Solution of triangles
html
local
div
role class
note hatnote navigation-not-searchable
html
local
a
href title
/wiki/Trigonometry Trigonometry
html
local
a
href title
/wiki/Plane_(geometry) Plane (geometry)
html
local
a
href title
/wiki/Sphere Sphere
html
local
a
href title
/wiki/Geodesy Geodesy
html
local
a
href title
/wiki/Astronomy Astronomy
html
local
a
href title
/wiki/Construction Construction
html
local
a
href title
/wiki/Navigation Navigation
html
local
p
html
local
span
class id
mw-headline Computing_the_area_of_a_triangle
html
local
h2
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Triangle.GeometryArea.svg/300px-Triangle.GeometryArea.svg.png 300 75 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Triangle.GeometryArea.svg/450px-Triangle.GeometryArea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Triangle.GeometryArea.svg/600px-Triangle.GeometryArea.svg.png 2x 504 126
html
local
a
href class
/wiki/File:Triangle.GeometryArea.svg image
html
local
a
href class title
/wiki/File:Triangle.GeometryArea.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href title
/wiki/Parallelogram Parallelogram
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:302px;
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}bh}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd11bbb2f3467193cdc32350cbdbc3d76930e67 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:9.07ex; height:5.176ex; T={\frac {1}{2}}bh
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
i
html
local
a
href title
/wiki/Aryabhata Aryabhata
html
local
a
href title
/wiki/Mathematician Mathematician
html
local
a
href title
/wiki/Astronomer Astronomer
html
local
a
href title
/wiki/Indian_mathematics Indian mathematics
html
local
a
href title
/wiki/Indian_astronomy Indian astronomy
html
local
a
href title
/wiki/Aryabhatiya Aryabhatiya
html
local
i
html
local
p
html
local
a
href
#cite_note-14
html
local
sup
id class
cite_ref-14 reference
html
local
p
html
local
span
class id
mw-headline Using_trigonometry
html
local
h3
html
local
img
alt src width height class data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Triangle.TrigArea.svg/165px-Triangle.TrigArea.svg.png 165 148 thumbimage 165 148
html
local
a
href class
/wiki/File:Triangle.TrigArea.svg image
html
local
i
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:167px;
html
local
a
href title
/wiki/Trigonometry Trigonometry
html
local
p
html
local
i
html
local
i
html
local
i
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \gamma }
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:1.262ex; height:2.176ex; \gamma 
html
local
span
class
mwe-math-element
html
local
span
class
nowrap
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}bh}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd11bbb2f3467193cdc32350cbdbc3d76930e67 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:9.07ex; height:5.176ex; T={\frac {1}{2}}bh
html
local
span
class
mwe-math-element
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}ab\sin \gamma ={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta }
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9312453769dc962b9d832d2ee85a4d1ba4803 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:38.367ex; height:5.176ex; T={\frac {1}{2}}ab\sin \gamma ={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta 
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
i
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \gamma }
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:1.262ex; height:2.176ex; \gamma 
html
local
span
class
mwe-math-element
html
local
i
html
local
i
html
local
b
html
local
p
html
local
i
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \gamma }
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:1.262ex; height:2.176ex; \gamma 
html
local
span
class
mwe-math-element
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}ab\sin(\alpha +\beta )={\frac {1}{2}}bc\sin(\beta +\gamma )={\frac {1}{2}}ca\sin(\gamma +\alpha ).}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4f11120a2c6a2af01b17a780ad167d8d4f6622 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:55.884ex; height:5.176ex; T={\frac {1}{2}}ab\sin(\alpha +\beta )={\frac {1}{2}}bc\sin(\beta +\gamma )={\frac {1}{2}}ca\sin(\gamma +\alpha ).
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mrow
html
local
mn
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {b^{2}(\sin \alpha )(\sin(\alpha +\beta ))}{2\sin \beta }},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/08a04d86731fac5b3901135398707478bb8f965e mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:26.943ex; height:6.343ex; T={\frac {b^{2}(\sin \alpha )(\sin(\alpha +\beta ))}{2\sin \beta }},
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
i
html
local
p
html
local
i
html
local
a
href
#cite_note-15
html
local
sup
id class
cite_ref-15 reference
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mn
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mn
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {a^{2}}{2(\cot \beta +\cot \gamma )}}={\frac {a^{2}(\sin \beta )(\sin \gamma )}{2\sin(\beta +\gamma )}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/4178692c97f1f01c509963467aaeaf534c9fbbb0 mwe-math-fallback-image-inline true vertical-align: -2.671ex; width:40.514ex; height:6.676ex; T={\frac {a^{2}}{2(\cot \beta +\cot \gamma )}}={\frac {a^{2}(\sin \beta )(\sin \gamma )}{2\sin(\beta +\gamma )}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
i
html
local
p
html
local
span
id
Using_Heron.27s_formula
html
local
span
class id
mw-headline Using_Heron's_formula
html
local
h3
html
local
a
href title
/wiki/Heron%27s_formula Heron's formula
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\sqrt {s(s-a)(s-b)(s-c)}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/1c17aa0115320b3e0cddc5c38aaccd0e9bc7ddc8 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:28.604ex; height:4.843ex; T={\sqrt {s(s-a)(s-b)(s-c)}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mn
html
local
mfrac
html
local
mstyle
displaystyle scriptlevel
false 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle s={\tfrac {a+b+c}{2}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/21dbbfafb813cea25e7ceeee5ea3068c36e9e104 mwe-math-fallback-image-inline true vertical-align: -1.171ex; width:9.869ex; height:3.676ex; s={\tfrac {a+b+c}{2}}
html
local
span
class
mwe-math-element
html
local
a
href title
/wiki/Semiperimeter Semiperimeter
html
local
p
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mo
stretchy
false
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{4}}{\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/3504cd6eebc34399e9c33a22540c9b9f366fd6fe mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:41.888ex; height:5.176ex; T={\frac {1}{4}}{\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mo
stretchy
false
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c92276984fa240ec74bc191b71e109d3ed17d5 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:47.231ex; height:5.176ex; T={\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{4}}{\sqrt {(a+b-c)(a-b+c)(-a+b+c)(a+b+c)}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/796b66a0ff5ea8f8c52beb840fbbb46ccd68392b mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:54.409ex; height:5.176ex; T={\frac {1}{4}}{\sqrt {(a+b-c)(a-b+c)(-a+b+c)(a+b+c)}}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Using_vectors
html
local
h3
html
local
a
href title
/wiki/Parallelogram Parallelogram
html
local
a
href title
/wiki/Euclidean_space Euclidean space
html
local
a
href class title
/wiki/Vector_(geometric) mw-redirect Vector (geometric)
html
local
b
html
local
b
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
p
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle |\mathbf {AB} \times \mathbf {AC} |,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/002c3e53ffe19f5bdc870ae1614eec6e3fdd7c32 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:12.652ex; height:2.843ex; |\mathbf {AB} \times \mathbf {AC} |,
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
a
href title
/wiki/Cross_product Cross product
html
local
b
html
local
b
html
local
p
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {1}{2}}|\mathbf {AB} \times \mathbf {AC} |.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/c09205b6d90612a48017f13f6651e69d8b00d982 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:14.651ex; height:5.176ex; {\frac {1}{2}}|\mathbf {AB} \times \mathbf {AC} |.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
a
href title
/wiki/Dot_product Dot product
html
local
p
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mo
html
local
mo
stretchy
false
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
stretchy
false
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
mathvariant
bold
html
local
mi
mathvariant
bold
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mspace
width
thinmathspace
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}.\,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e671b1782de42eafdb90f012a04ac7376c84ca mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:78.572ex; height:5.176ex; {\frac {1}{2}}{\sqrt {(\mathbf {AB} \cdot \mathbf {AB} )(\mathbf {AC} \cdot \mathbf {AC} )-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}={\frac {1}{2}}{\sqrt {|\mathbf {AB} |^{2}|\mathbf {AC} |^{2}-(\mathbf {AB} \cdot \mathbf {AC} )^{2}}}.\,
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
b
html
local
a
href title
/wiki/Euclidean_vector#In_Cartesian_space Euclidean vector
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
b
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
p
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mspace
width
thinmathspace
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mspace
width
thinmathspace
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {1}{2}}\,|x_{1}y_{2}-x_{2}y_{1}|.\,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/5886b6b1f4150bf50a5ace37046516beab8bcee0 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:16.708ex; height:5.176ex; {\frac {1}{2}}\,|x_{1}y_{2}-x_{2}y_{1}|.\,
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Using_coordinates
html
local
h3
html
local
i
html
local
a
href title
/wiki/Cartesian_coordinate_system Cartesian coordinate system
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
sup
html
local
sub
html
local
span
class
frac nowrap
html
local
a
href title
/wiki/Absolute_value Absolute value
html
local
a
href title
/wiki/Determinant Determinant
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mo
movablelimits form
true prefix
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mtable
rowspacing columnspacing
4pt 1em
html
local
mo
html
local
mrow
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mo
html
local
mrow
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b723f220e96ded1a725573473ee4dea6e2e411 mwe-math-fallback-image-inline true vertical-align: -2.505ex; width:44.884ex; height:6.176ex; T={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{pmatrix}}\right|={\frac {1}{2}}|x_{B}y_{C}-x_{C}y_{B}|.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mo
movablelimits form
true prefix
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mtr
html
local
mtable
rowspacing columnspacing
4pt 1em
html
local
mo
html
local
mrow
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mo
html
local
mrow
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
maxsize minsize
1.2em 1.2em
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
maxsize minsize
1.2em 1.2em
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{B}-x_{A}y_{C}+x_{B}y_{C}-x_{B}y_{A}+x_{C}y_{A}-x_{C}y_{B}{\big |},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ac319c9aabab109fbe264a87644182c36cd1c2 mwe-math-fallback-image-inline true vertical-align: -4.171ex; width:83.665ex; height:9.509ex; T={\frac {1}{2}}\left|\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right|={\frac {1}{2}}{\big |}x_{A}y_{B}-x_{A}y_{C}+x_{B}y_{C}-x_{B}y_{A}+x_{C}y_{A}-x_{C}y_{B}{\big |},
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
maxsize minsize
1.2em 1.2em
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mo
maxsize minsize
1.2em 1.2em
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}{\big |}(x_{A}-x_{C})(y_{B}-y_{A})-(x_{A}-x_{B})(y_{C}-y_{A}){\big |}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/206007a4d7be5d914a8540a19d5fa80a566e76d1 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:51.77ex; height:5.176ex; T={\frac {1}{2}}{\big |}(x_{A}-x_{C})(y_{B}-y_{A})-(x_{A}-x_{B})(y_{C}-y_{A}){\big |}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
a
href
#cite_note-16
html
local
sup
id class
cite_ref-16 reference
html
local
a
href title
/wiki/Shoelace_formula Shoelace formula
html
local
p
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
mi
html
local
mo
stretchy
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\bar {a}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/6513ea66e4b94de05c92bc66f00e6e021d1c2a9c mwe-math-fallback-image-inline true vertical-align: -0.338ex; width:1.23ex; height:2.009ex; {\bar {a}}
html
local
span
class
mwe-math-element
html
local
mi
html
local
mo
stretchy
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\bar {b}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/89e44ed80d87353f07b93aa5b68876724e564ff1 mwe-math-fallback-image-inline true vertical-align: -0.338ex; width:1.162ex; height:2.509ex; {\bar {b}}
html
local
span
class
mwe-math-element
html
local
mi
html
local
mo
stretchy
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\bar {c}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/94e1e99531b83728c7362b99dda09cc8b366168e mwe-math-fallback-image-inline true vertical-align: -0.338ex; width:1.292ex; height:2.009ex; {\bar {c}}
html
local
span
class
mwe-math-element
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mtd
html
local
mi
html
local
mo
stretchy
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mtd
html
local
mn
html
local
mtd
html
local
mtr
html
local
mi
html
local
mtd
html
local
mi
html
local
mo
stretchy
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mtd
html
local
mn
html
local
mtd
html
local
mtr
html
local
mi
html
local
mtd
html
local
mi
html
local
mo
stretchy
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mtd
html
local
mn
html
local
mtd
html
local
mtr
html
local
mtable
rowspacing columnspacing
4pt 1em
html
local
mo
html
local
mrow
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {i}{4}}{\begin{vmatrix}a&{\bar {a}}&1\\b&{\bar {b}}&1\\c&{\bar {c}}&1\end{vmatrix}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/f828eba6cac415431233abcf5b263b861d8e1efd mwe-math-fallback-image-inline true vertical-align: -4.171ex; width:17.108ex; height:9.509ex; T={\frac {i}{4}}{\begin{vmatrix}a&{\bar {a}}&1\\b&{\bar {b}}&1\\c&{\bar {c}}&1\end{vmatrix}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
a
href class title
/wiki/Pythagorean_sum mw-redirect Pythagorean sum
html
local
i
html
local
i
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mtr
html
local
mtable
rowspacing columnspacing
4pt 1em
html
local
mo
html
local
mrow
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mtr
html
local
mtable
rowspacing columnspacing
4pt 1em
html
local
mo
html
local
mrow
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mtd
html
local
mtr
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mn
html
local
mtd
html
local
mtr
html
local
mtable
rowspacing columnspacing
4pt 1em
html
local
mo
html
local
mrow
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}{\sqrt {{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{vmatrix}}^{2}}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/67c599953dad11fdc117c9ecf5ad8c56de3563e0 mwe-math-fallback-image-inline true vertical-align: -4.171ex; width:63.421ex; height:10.509ex; T={\frac {1}{2}}{\sqrt {{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{vmatrix}}^{2}}}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Using_line_integrals
html
local
h3
html
local
a
href title
/wiki/Line_integral Line integral
html
local
i
html
local
i
html
local
i
html
local
i
html
local
p
html
local
a
href title
/wiki/Polygon Polygon
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
sub
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
span
class
nowrap
html
local
p
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
p
html
local
a
href class title
/wiki/Polar_coordinates mw-redirect Polar coordinates
html
local
a
href class title
/wiki/Cartesian_coordinates mw-redirect Cartesian coordinates
html
local
sub
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
sub
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
span
class
nowrap
html
local
i
html
local
i
html
local
span
class
nowrap
html
local
span
class
nowrap
html
local
p
html
local
span
id
Formulas_resembling_Heron.27s_formula
html
local
span
class id
mw-headline Formulas_resembling_Heron's_formula
html
local
h3
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
span
class
nowrap
html
local
a
href
#cite_note-17
html
local
sup
id class
cite_ref-17 reference
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {4}{3}}{\sqrt {\sigma (\sigma -m_{a})(\sigma -m_{b})(\sigma -m_{c})}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/e476af67eb23797667d2283ddcb9235618640475 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:38.077ex; height:5.176ex; T={\frac {4}{3}}{\sqrt {\sigma (\sigma -m_{a})(\sigma -m_{b})(\sigma -m_{c})}}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
stretchy
false
html
local
mo
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle H=(h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1})/2}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/79dcc8db8859bf4b23d773e7b64748223f4f76d9 mwe-math-fallback-image-inline true vertical-align: -1.005ex; width:25.992ex; height:3.343ex; H=(h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1})/2
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-18
html
local
sup
id class
cite_ref-18 reference
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T^{-1}=4{\sqrt {H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/1c09032d3d1d94594db96bf9279e7c940b0662f8 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:44.503ex; height:4.843ex; T^{-1}=4{\sqrt {H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
span
class
nowrap
html
local
a
href
#cite_note-19
html
local
sup
id class
cite_ref-19 reference
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T=D^{2}{\sqrt {S(S-\sin \alpha )(S-\sin \beta )(S-\sin \gamma )}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a5ea9fea9519ecf0e8cd1c333319c03cf54a8f mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:43.793ex; height:4.843ex; T=D^{2}{\sqrt {S(S-\sin \alpha )(S-\sin \beta )(S-\sin \gamma )}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mstyle
displaystyle scriptlevel
false 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mstyle
displaystyle scriptlevel
false 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mstyle
displaystyle scriptlevel
false 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle D={\tfrac {a}{\sin \alpha }}={\tfrac {b}{\sin \beta }}={\tfrac {c}{\sin \gamma }}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/57e427b51aff94e59dc5a32fb7283314fe312c3b mwe-math-fallback-image-inline true vertical-align: -1.671ex; width:24.48ex; height:4.176ex; D={\tfrac {a}{\sin \alpha }}={\tfrac {b}{\sin \beta }}={\tfrac {c}{\sin \gamma }}.
html
local
span
class
mwe-math-element
html
local
p
html
local
span
id
Using_Pick.27s_theorem
html
local
span
id
Using_Pick's_Theorem
html
local
span
class id
mw-headline Using_Pick's_theorem
html
local
h3
html
local
a
href title
/wiki/Pick%27s_theorem Pick's theorem
html
local
a
href title
/wiki/Lattice_graph Lattice graph
html
local
p
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mn
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T=I+{\frac {1}{2}}B-1}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1194e0024003c447803126fe0e47d055d53cfd mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:16.512ex; height:5.176ex; T=I+{\frac {1}{2}}B-1
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle I}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f mwe-math-fallback-image-inline true vertical-align: -0.338ex; width:1.172ex; height:2.176ex; I
html
local
span
class
mwe-math-element
html
local
i
html
local
i
html
local
p
html
local
span
class id
mw-headline Other_area_formulas
html
local
h3
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T=r\cdot s,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/aab6ed54484e772842a29d7887a1bf4248da0b09 mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:9.2ex; height:2.509ex; T=r\cdot s,
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
a
href class title
/wiki/Inradius mw-redirect Inradius
html
local
i
html
local
a
href title
/wiki/Semiperimeter Semiperimeter
html
local
i
html
local
a
href title
/wiki/Tangential_polygon Tangential polygon
html
local
a
href
#cite_note-Kiss-20
html
local
sup
id class
cite_ref-Kiss_20-0 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T=r_{a}(s-a)=r_{b}(s-b)=r_{c}(s-c)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9db8098165e2df1828c8f4aa9dee93791350b3 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:37.516ex; height:2.843ex; {\displaystyle T=r_{a}(s-a)=r_{b}(s-b)=r_{c}(s-c)}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mspace
width
thinmathspace
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mspace
width
thinmathspace
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle r_{a},\,r_{b},\,r_{c}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/d4bae033391ead78c38d3475f86aacacdc2c2c80 mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:8.972ex; height:2.009ex; {\displaystyle r_{a},\,r_{b},\,r_{c}}
html
local
span
class
mwe-math-element
html
local
a
href class title
/wiki/Excircles mw-redirect Excircles
html
local
i
html
local
p
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}D^{2}(\sin \alpha )(\sin \beta )(\sin \gamma )}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/d18be8e91139f54457dbd3f50ced879beb19fb54 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:28.95ex; height:5.176ex; T={\frac {1}{2}}D^{2}(\sin \alpha )(\sin \beta )(\sin \gamma )
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
a
href
#cite_note-21
html
local
sup
id class
cite_ref-21 reference
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
html
local
mn
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
html
local
mn
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {abc}{2D}}={\frac {abc}{4R}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce6f47781df12cae15b429b22cd50a333a7e4b9 mwe-math-fallback-image-inline true vertical-align: -2.005ex; width:15.974ex; height:5.509ex; T={\frac {abc}{2D}}={\frac {abc}{4R}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
a
href
#cite_note-22
html
local
sup
id class
cite_ref-22 reference
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {\tan \alpha }{4}}(b^{2}+c^{2}-a^{2})}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/80ef19e6cc7cd350d2494deb45298c36d30c6bbb mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:24.692ex; height:5.176ex; T={\frac {\tan \alpha }{4}}(b^{2}+c^{2}-a^{2})
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
a
href
#cite_note-23
html
local
sup
id class
cite_ref-23 reference
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\sqrt {rr_{a}r_{b}r_{c}}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/2991578701fc926564871abc041dadfadee64517 mwe-math-fallback-image-inline true vertical-align: -1.171ex; width:14.496ex; height:3.009ex; {\displaystyle T={\sqrt {rr_{a}r_{b}r_{c}}}.}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
a
href
#cite_note-24
html
local
sup
id class
cite_ref-24 reference
html
local
p
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
stretchy
false
html
local
mn
html
local
mo
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}[abch_{a}h_{b}h_{c}]^{1/3},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/2a42df38ee19d6fdcb63123fceb86327412c60e5 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:21.607ex; height:5.176ex; T={\frac {1}{2}}[abch_{a}h_{b}h_{c}]^{1/3},
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {1}{2}}{\sqrt {abh_{a}h_{b}}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/3664554c2fa205a533b4d79564fdf8a786c0a992 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:16.649ex; height:5.176ex; T={\frac {1}{2}}{\sqrt {abh_{a}h_{b}}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mn
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
stretchy
false
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {a+b}{2(h_{a}^{-1}+h_{b}^{-1})}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/5536ea4679ab06fb032f31ed91b809a6657028f2 mwe-math-fallback-image-inline true vertical-align: -3.171ex; width:19.373ex; height:6.676ex; T={\frac {a+b}{2(h_{a}^{-1}+h_{b}^{-1})}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mrow
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {Rh_{b}h_{c}}{a}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/8de3a914bc4862a31e9183521df52ef9ad3c9d91 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:11.895ex; height:5.343ex; T={\frac {Rh_{b}h_{c}}{a}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mrow
html
local
mn
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T={\frac {h_{a}h_{b}}{2\sin \gamma }}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/bea395c9fcb3505acbc4ca2d4444198dbd5e0b5e mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:12.272ex; height:5.843ex; T={\frac {h_{a}h_{b}}{2\sin \gamma }}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Upper_bound_on_the_area
html
local
h3
html
local
i
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mn
html
local
mn
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mfrac
html
local
mstyle
displaystyle scriptlevel
false 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T\leq {\tfrac {p^{2}}{12{\sqrt {3}}}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/300f28cd9dc99b75a3d961630071fbee8194208d mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:10.053ex; height:5.009ex; {\displaystyle T\leq {\tfrac {p^{2}}{12{\sqrt {3}}}},}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
a
href
#cite_note-25
html
local
sup
id class
cite_ref-25 reference
html
local
a
href
#cite_note-26
html
local
sup
id class
cite_ref-26 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
i
html
local
a
href
#cite_note-27
html
local
sup
id class
cite_ref-27 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mn
html
local
mn
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle 4{\sqrt {3}}T\leq a^{2}+b^{2}+c^{2}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d69705964f836b00dd802cbd63bf08f4c59c93 mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:21.073ex; height:3.009ex; 4{\sqrt {3}}T\leq a^{2}+b^{2}+c^{2}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mn
html
local
mn
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle 4{\sqrt {3}}T\leq {\frac {9abc}{a+b+c}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/cad8cdcc5f5349abbfbf200eae66829704b30d6a mwe-math-fallback-image-inline true vertical-align: -2.171ex; width:19.393ex; height:5.676ex; 4{\sqrt {3}}T\leq {\frac {9abc}{a+b+c}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
span
class id
mw-headline Bisecting_the_area
html
local
h3
html
local
a
href title
/wiki/Bisection#Area_bisectors_and_perimeter_bisectors Bisection
html
local
a
href
#cite_note-Dunn-28
html
local
sup
id class
cite_ref-Dunn_28-0 reference
html
local
p
html
local
p
html
local
span
class id
mw-headline Further_formulas_for_general_Euclidean_triangles
html
local
h2
html
local
a
href title
/wiki/List_of_triangle_inequalities List of triangle inequalities
html
local
div
role class
note hatnote navigation-not-searchable
html
local
p
html
local
span
id
Medians.2C_angle_bisectors.2C_perpendicular_side_bisectors.2C_and_altitudes
html
local
span
class id
mw-headline Medians,_angle_bisectors,_perpendicular_side_bisectors,_and_altitudes
html
local
h3
html
local
a
href title
/wiki/Median_(geometry) Median (geometry)
html
local
a
href class title
/wiki/Angle_bisector mw-redirect Angle bisector
html
local
a
href title
/wiki/Bisection#Perpendicular_bisectors Bisection
html
local
a
href class title
/wiki/Altitude_(geometry) mw-redirect Altitude (geometry)
html
local
div
role class
note hatnote navigation-not-searchable
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-0 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msubsup
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {3}{4}}(a^{2}+b^{2}+c^{2})=m_{a}^{2}+m_{b}^{2}+m_{c}^{2}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/95c9bf0f661b70d8b7a3e6e1aac4bf92e56225a7 mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:33.996ex; height:5.176ex; {\frac {3}{4}}(a^{2}+b^{2}+c^{2})=m_{a}^{2}+m_{b}^{2}+m_{c}^{2}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
stretchy
false
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle m_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\sqrt {{\frac {1}{2}}(a^{2}+b^{2}+c^{2})-{\frac {3}{4}}a^{2}}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/15bd5f29a8c2112afbc1f794dc3f462f2549c77f mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:53.397ex; height:6.176ex; m_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\sqrt {{\frac {1}{2}}(a^{2}+b^{2}+c^{2})-{\frac {3}{4}}a^{2}}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
p
html
local
i
html
local
a
href
#cite_note-30
html
local
sup
id class
cite_ref-30 reference
html
local
p
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mn
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mo
html
local
mrow
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle w_{A}={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}={\sqrt {bc\left[1-{\frac {a^{2}}{(b+c)^{2}}}\right]}}={\frac {2bc}{b+c}}\cos {\frac {A}{2}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/0161030c29669008d351fef1242b8b8e4fc791db mwe-math-fallback-image-inline true vertical-align: -3.338ex; width:59.321ex; height:7.676ex; {\displaystyle w_{A}={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}={\sqrt {bc\left[1-{\frac {a^{2}}{(b+c)^{2}}}\right]}}={\frac {2bc}{b+c}}\cos {\frac {A}{2}},}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
p
html
local
p
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle p_{a}={\frac {2aT}{a^{2}+b^{2}-c^{2}}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/7758ec210beadbdc2c5ba14211f176b53318f188 mwe-math-fallback-image-inline true vertical-align: -2.338ex; margin-left: -0.089ex; width:19.02ex; height:5.676ex; p_{a}={\frac {2aT}{a^{2}+b^{2}-c^{2}}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle p_{b}={\frac {2bT}{a^{2}+b^{2}-c^{2}}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/67f251cab5b3866713624712dcc59aae48c54589 mwe-math-fallback-image-inline true vertical-align: -2.338ex; margin-left: -0.089ex; width:18.856ex; height:5.843ex; p_{b}={\frac {2bT}{a^{2}+b^{2}-c^{2}}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle p_{c}={\frac {2cT}{a^{2}-b^{2}+c^{2}}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/89fa4a3ac80de2f3f0547ac5136403ea12872b8f mwe-math-fallback-image-inline true vertical-align: -2.338ex; margin-left: -0.089ex; width:18.862ex; height:5.676ex; p_{c}={\frac {2cT}{a^{2}-b^{2}+c^{2}}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle a\geq b\geq c}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/8a82e6284fedb32392e7cd3a5d0284907898bb0a mwe-math-fallback-image-inline true vertical-align: -0.505ex; width:9.431ex; height:2.343ex; a\geq b\geq c
html
local
span
class
mwe-math-element
html
local
mi
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle T.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/4de28b735beca1303bc5da9ba518a5a22a70a5d4 mwe-math-fallback-image-inline true vertical-align: -0.338ex; width:2.283ex; height:2.176ex; T.
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-31
html
local
sup
id class
cite_ref-31 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
i
html
local
p
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mn
html
local
mi
html
local
mrow
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle h_{a}={\frac {2T}{a}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb8623f199fecea05be6cb83516ac19f4fce0cc mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:9.821ex; height:5.176ex; h_{a}={\frac {2T}{a}}.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Circumradius_and_inradius
html
local
h3
html
local
a
href class title
/wiki/Circumradius mw-redirect Circumradius
html
local
a
href class title
/wiki/Inradius mw-redirect Inradius
html
local
div
role class
note hatnote navigation-not-searchable
html
local
i
html
local
i
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mfrac
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle R={\sqrt {\frac {a^{2}b^{2}c^{2}}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}};}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/a2363cceabee8193590520a80a3225e855cd7ed1 mwe-math-fallback-image-inline true vertical-align: -3.338ex; width:53.375ex; height:7.676ex; R={\sqrt {\frac {a^{2}b^{2}c^{2}}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}};
html
local
span
class
mwe-math-element
html
local
dd
html
local
mi
html
local
mo
html
local
mo
stretchy
false
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mn
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mfrac
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle r={\sqrt {\frac {(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)}}};}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/1f2444976112e512b8c9a0c62187e4016ee95eda mwe-math-fallback-image-inline true vertical-align: -3.171ex; width:41.935ex; height:7.509ex; r={\sqrt {\frac {(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)}}};
html
local
span
class
mwe-math-element
html
local
dd
html
local
mn
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {1}{r}}={\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/053bcbf085e893a90fa9ba0a6ace801b1bbd6d48 mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:20.287ex; height:5.676ex; {\frac {1}{r}}={\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
sub
html
local
i
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-1 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mi
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mrow
html
local
mi
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {r}{R}}={\frac {4T^{2}}{sabc}}=\cos \alpha +\cos \beta +\cos \gamma -1;}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/1b48e3aa464e3eb8575464826867356d93e55a25 mwe-math-fallback-image-inline true vertical-align: -2.005ex; width:38.865ex; height:5.843ex; {\frac {r}{R}}={\frac {4T^{2}}{sabc}}=\cos \alpha +\cos \beta +\cos \gamma -1;
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-LH-11
html
local
sup
id class
cite_ref-LH_11-1 reference
html
local
dd
html
local
dl
html
local
p
html
local
mn
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle 2Rr={\frac {abc}{a+b+c}}}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/663e37165bc07debf90f77989138647fd781e288 mwe-math-fallback-image-inline true vertical-align: -2.171ex; width:16.825ex; height:5.676ex; 2Rr={\frac {abc}{a+b+c}}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-2 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mo
html
local
mspace
width
1em
html
local
mspace
width
1em
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mo
html
local
mspace
width
1em
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle ab=h_{c}D,\quad \quad bc=h_{a}D,\quad ca=h_{b}D.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb9e087af3d4892df17a9fe650a35b2338ec80c mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:38.22ex; height:2.509ex; {\displaystyle ab=h_{c}D,\quad \quad bc=h_{a}D,\quad ca=h_{b}D.}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Adjacent_triangles
html
local
h3
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
a
href title
/wiki/Cyclic_quadrilateral Cyclic quadrilateral
html
local
i
html
local
i
html
local
i
html
local
i
html
local
a
href
#cite_note-Johnson-32
html
local
sup
id class
cite_ref-Johnson_32-0 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mspace
width
thinmathspace
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle f^{2}={\frac {(ac+bd)(ad+bc)}{(ab+cd)}}.\,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/42335dac78e451a318e8893f1862b965564373d1 mwe-math-fallback-image-inline true vertical-align: -2.671ex; width:25.543ex; height:6.509ex; f^{2}={\frac {(ac+bd)(ad+bc)}{(ab+cd)}}.\,
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Centroid
html
local
h3
html
local
a
href title
/wiki/Centroid Centroid
html
local
div
role class
note hatnote navigation-not-searchable
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
a
href
#cite_note-Johnson-32
html
local
sup
id class
cite_ref-Johnson_32-1 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mo
stretchy
false
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mspace
width
thinmathspace
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle (PA)^{2}+(PB)^{2}+(PC)^{2}=(GA)^{2}+(GB)^{2}+(GC)^{2}+3(PG)^{2}.\,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/677e8e9762b14a38b1b55cad61993cad8661e097 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:64.377ex; height:3.176ex; (PA)^{2}+(PB)^{2}+(PC)^{2}=(GA)^{2}+(GB)^{2}+(GC)^{2}+3(PG)^{2}.\,
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mo
stretchy
false
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
stretchy
false
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle AB^{2}+BC^{2}+CA^{2}=3(GA^{2}+GB^{2}+GC^{2}).}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/e0813f34f24908b350125c0ce9065bd1155847a3 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:45.767ex; height:3.176ex; AB^{2}+BC^{2}+CA^{2}=3(GA^{2}+GB^{2}+GC^{2}).
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-33
html
local
sup
id class
cite_ref-33 reference
html
local
dd
html
local
dl
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
i
html
local
i
html
local
a
href
#cite_note-Johnson-32
html
local
sup
id class
cite_ref-Johnson_32-2 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mspace
width
1em
html
local
mspace
width
1em
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mspace
width
1em
html
local
mspace
width
1em
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mspace
width
thinmathspace
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {q_{a}}{q_{b}}}={\frac {b}{a}},\quad \quad {\frac {q_{b}}{q_{c}}}={\frac {c}{b}},\quad \quad {\frac {q_{a}}{q_{c}}}={\frac {c}{a}}\,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/349bb9fbaac0ef7865edba16623af5f921df078e mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:35.783ex; height:5.843ex; {\displaystyle {\frac {q_{a}}{q_{b}}}={\frac {b}{a}},\quad \quad {\frac {q_{b}}{q_{c}}}={\frac {c}{b}},\quad \quad {\frac {q_{a}}{q_{c}}}={\frac {c}{a}}\,}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mi
html
local
mo
html
local
mn
html
local
mn
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mi
html
local
mspace
width
thinmathspace
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle q_{a}\cdot a=q_{b}\cdot b=q_{c}\cdot c={\frac {2}{3}}T\,}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/b823c678c3a8cd3d8d70e005395c4a3c1d54f4bc mwe-math-fallback-image-inline true vertical-align: -1.838ex; width:27.684ex; height:5.176ex; {\displaystyle q_{a}\cdot a=q_{b}\cdot b=q_{c}\cdot c={\frac {2}{3}}T\,}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
p
html
local
span
id
Circumcenter.2C_incenter.2C_and_orthocenter
html
local
span
class id
mw-headline Circumcenter,_incenter,_and_orthocenter
html
local
h3
html
local
a
href class title
/wiki/Circumcenter mw-redirect Circumcenter
html
local
a
href title
/wiki/Incenter Incenter
html
local
a
href class title
/wiki/Orthocenter mw-redirect Orthocenter
html
local
div
role class
note hatnote navigation-not-searchable
html
local
a
href class title
/wiki/Carnot%27s_Theorem mw-redirect Carnot's Theorem
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-3 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
a
href title
/wiki/Lie_algebra Lie algebra
html
local
p
html
local
a
href title
/wiki/Euler%27s_theorem_in_geometry Euler's theorem in geometry
html
local
i
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-4 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mo
stretchy
false
html
local
mi
html
local
mo
html
local
mn
html
local
mi
html
local
mo
stretchy
false
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \displaystyle d^{2}=R(R-2r)}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3dc80966816a188b1ec4c8747bdded2b4e6726 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:15.76ex; height:3.176ex; \displaystyle d^{2}=R(R-2r)
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
p
html
local
mn
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mo
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mi
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fcc54cdab2027694aba87f9806229d6bfb686d mwe-math-fallback-image-inline true vertical-align: -2.171ex; width:21.897ex; height:5.509ex; {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
i
html
local
i
html
local
i
html
local
p
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
i
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-5 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-6 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
i
html
local
a
href
#cite_note-Altshiller-Court-29
html
local
sup
id class
cite_ref-Altshiller-Court_29-7 reference
html
local
span
html
local
sup
class style
reference white-space:nowrap;
html
local
p
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle AH^{2}+BH^{2}+CH^{2}+a^{2}+b^{2}+c^{2}=12R^{2}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/d491c46b36a59f8ec4ba69f738b76eb6bd61bb26 mwe-math-fallback-image-inline true vertical-align: -0.505ex; width:44.234ex; height:2.843ex; {\displaystyle AH^{2}+BH^{2}+CH^{2}+a^{2}+b^{2}+c^{2}=12R^{2}.}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Angles
html
local
h3
html
local
a
href title
/wiki/Law_of_sines Law of sines
html
local
a
href title
/wiki/Law_of_cosines Law of cosines
html
local
a
href title
/wiki/Law_of_tangents Law of tangents
html
local
a
href
#Trigonometric_conditions
html
local
p
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mspace
width
1em
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mspace
width
1em
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle a=b\cos C+c\cos B,\quad b=c\cos A+a\cos C,\quad c=a\cos B+b\cos A.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/c7276aa21cbaefaea5bd112fb30b5d870cc9faa0 mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:68.738ex; height:2.509ex; a=b\cos C+c\cos B,\quad b=c\cos A+a\cos C,\quad c=a\cos B+b\cos A.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
id
Morley.27s_trisector_theorem
html
local
span
class id
mw-headline Morley's_trisector_theorem
html
local
h2
html
local
a
href title
/wiki/Morley%27s_trisector_theorem Morley's trisector theorem
html
local
div
role class
note hatnote navigation-not-searchable
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Morley_triangle.svg/220px-Morley_triangle.svg.png 220 147 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Morley_triangle.svg/330px-Morley_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Morley_triangle.svg/440px-Morley_triangle.svg.png 2x 450 300
html
local
a
href class
/wiki/File:Morley_triangle.svg image
html
local
a
href class title
/wiki/File:Morley_triangle.svg internal Enlarge
html
local
div
class
magnify
html
local
a
href title
/wiki/Finite_subdivision_rule Finite subdivision rule
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:222px;
html
local
a
href class title
/wiki/Angle_trisector mw-redirect Angle trisector
html
local
p
html
local
span
class id
mw-headline Figures_inscribed_in_a_triangle
html
local
h2
html
local
span
class id
mw-headline Conics
html
local
h3
html
local
p
html
local
a
href title
/wiki/Steiner_inellipse Steiner inellipse
html
local
a
href title
/wiki/Marden%27s_theorem Marden's theorem
html
local
a
href title
/wiki/Ellipse#Elements_of_an_ellipse Ellipse
html
local
a
href
#cite_note-34
html
local
sup
id class
cite_ref-34 reference
html
local
p
html
local
a
href title
/wiki/Mandart_inellipse Mandart inellipse
html
local
p
html
local
i
html
local
i
html
local
i
html
local
a
href
#cite_note-35
html
local
sup
id class
cite_ref-35 reference
html
local
p
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
html
local
mo
accent
false
html
local
mover
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mn
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa55cb2a4b60d1ecae688927cd4f140a5518e91 mwe-math-fallback-image-inline true vertical-align: -2.838ex; width:40.258ex; height:7.176ex; {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
span
class id
mw-headline Convex_polygon
html
local
h3
html
local
i
html
local
i
html
local
a
href title
/wiki/Parallelogram Parallelogram
html
local
a
href
#cite_note-36
html
local
sup
id class
cite_ref-36 reference
html
local
p
html
local
span
class id
mw-headline Hexagon
html
local
h3
html
local
a
href title
/wiki/Lemoine_hexagon Lemoine hexagon
html
local
a
href title
/wiki/Hexagon#Cyclic_hexagon Hexagon
html
local
a
href class title
/wiki/Symmedian_point mw-redirect Symmedian point
html
local
a
href title
/wiki/Polygon#Convexity_and_types_of_non-convexity Polygon
html
local
p
html
local
span
class id
mw-headline Squares
html
local
h3
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
i
html
local
sub
html
local
i
html
local
i
html
local
a
href
#cite_note-37
html
local
sup
id class
cite_ref-37 reference
html
local
a
href
#cite_note-OS-38
html
local
sup
id class
cite_ref-OS_38-0 reference
html
local
p
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mo
html
local
mn
html
local
mi
html
local
mi
html
local
mrow
html
local
mi
html
local
mn
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msup
html
local
mo
html
local
mn
html
local
mi
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mi
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mrow
html
local
mi
html
local
mo
html
local
mi
html
local
mi
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
msub
html
local
mrow
html
local
mfrac
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6bfea611f3c1d6c4a694f9bc8f8e2055d4f542 mwe-math-fallback-image-inline true vertical-align: -2.338ex; width:25.089ex; height:5.843ex; {\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.}
html
local
span
class
mwe-math-element
html
local
dd
html
local
dl
html
local
i
html
local
sup
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
i
html
local
span
class
nowrap
html
local
i
html
local
i
html
local
mn
html
local
mn
html
local
msqrt
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mo
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
mn
html
local
mo
html
local
mn
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle 2{\sqrt {2}}/3=0.94....}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/c27b21e3de7dd4c4db125ee8b1f64b132cf6f3ab mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:16.406ex; height:3.176ex; 2{\sqrt {2}}/3=0.94....
html
local
span
class
mwe-math-element
html
local
a
href
#cite_note-OS-38
html
local
sup
id class
cite_ref-OS_38-1 reference
html
local
p
html
local
span
class id
mw-headline Triangles
html
local
h3
html
local
a
href title
/wiki/Pedal_triangle Pedal triangle
html
local
a
href class title
/wiki/Midpoint_triangle mw-redirect Midpoint triangle
html
local
p
html
local
a
href class title
/wiki/Gergonne_triangle mw-redirect Gergonne triangle
html
local
a
href title
/wiki/Extouch_triangle Extouch triangle
html
local
p
html
local
span
class id
mw-headline Figures_circumscribed_about_a_triangle
html
local
h2
html
local
a
href title
/wiki/Tangential_triangle Tangential triangle
html
local
a
href class title
/wiki/Tangent_line mw-redirect Tangent line
html
local
p
html
local
p
html
local
a
href title
/wiki/Steiner_ellipse Steiner ellipse
html
local
p
html
local
a
href class title
/wiki/Kiepert_hyperbola mw-redirect Kiepert hyperbola
html
local
a
href class title
/wiki/Conic mw-redirect Conic
html
local
p
html
local
a
href
#cite_note-39
html
local
sup
id class
cite_ref-39 reference
html
local
p
html
local
span
class id
mw-headline Specifying_the_location_of_a_point_in_a_triangle
html
local
h2
html
local
a
href class title
/wiki/Cartesian_plane mw-redirect Cartesian plane
html
local
p
html
local
p
html
local
a
href title
/wiki/Trilinear_coordinates Trilinear coordinates
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle x:y:z}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc1d4f62fa3da71416dec491efd083a8c473361 mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:7.448ex; height:2.009ex; x:y:z
html
local
span
class
mwe-math-element
html
local
mi
html
local
mo
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle x:y}
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/7c856cee7dfad676e081203a32e48077904001dc mwe-math-fallback-image-inline true vertical-align: -0.671ex; width:4.422ex; height:2.009ex; x:y
html
local
span
class
mwe-math-element
html
local
li
html
local
a
href class title
/wiki/Barycentric_coordinates_(mathematics) mw-redirect Barycentric coordinates (mathematics)
html
local
mi
html
local
mo
html
local
mi
html
local
mo
html
local
mi
html
local
mstyle
displaystyle scriptlevel
true 0
html
local
mrow
class
MJX-TeXAtom-ORD
html
local
annotation
encoding
application/x-tex
html
local
semantics
html
local
math
xmlns alttext
http://www.w3.org/1998/Math/MathML {\displaystyle \alpha :\beta :\gamma }
html
local
span
class style
mwe-math-mathml-inline mwe-math-mathml-a11y display: none;
html
local
img
src class aria-hidden style alt
https://wikimedia.org/api/rest_v1/media/math/render/svg/cdeb88b91752ebc42398e577d5455edc63aab124 mwe-math-fallback-image-inline true vertical-align: -0.838ex; width:7.956ex; height:2.676ex; \alpha :\beta :\gamma 
html
local
span
class
mwe-math-element
html
local
li
html
local
ul
html
local
span
class id
mw-headline Non-planar_triangles
html
local
h2
html
local
a
href class title
/wiki/Spherical_triangle mw-redirect Spherical triangle
html
local
a
href title
/wiki/Spherical_geometry Spherical geometry
html
local
a
href title
/wiki/Hyperbolic_triangle Hyperbolic triangle
html
local
a
href title
/wiki/Hyperbolic_geometry Hyperbolic geometry
html
local
p
html
local
a
href class title
/wiki/Saddle_surface mw-redirect Saddle surface
html
local
a
href title
/wiki/Sphere Sphere
html
local
a
href
#cite_note-40
html
local
sup
id class
cite_ref-40 reference
html
local
p
html
local
p
html
local
i
html
local
dd
html
local
dl
html
local
i
html
local
a
href title
/wiki/Great_circle Great circle
html
local
span
class
nowrap
html
local
i
html
local
span
class
nowrap
html
local
p
html
local
i
html
local
p
html
local
span
class id
mw-headline Triangles_in_construction
html
local
h2
html
local
a
href title
/wiki/Truss Truss
html
local
div
role class
note hatnote navigation-not-searchable
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Edificio_Fuller_%28Flatiron%29_en_2010_desde_el_Empire_State_crop_boxin.jpg/220px-Edificio_Fuller_%28Flatiron%29_en_2010_desde_el_Empire_State_crop_boxin.jpg 220 416 thumbimage //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Edificio_Fuller_%28Flatiron%29_en_2010_desde_el_Empire_State_crop_boxin.jpg/330px-Edificio_Fuller_%28Flatiron%29_en_2010_desde_el_Empire_State_crop_boxin.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Edificio_Fuller_%28Flatiron%29_en_2010_desde_el_Empire_State_crop_boxin.jpg/440px-Edificio_Fuller_%28Flatiron%29_en_2010_desde_el_Empire_State_crop_boxin.jpg 2x 1489 2818
html
local
a
href class
/wiki/File:Edificio_Fuller_(Flatiron)_en_2010_desde_el_Empire_State_crop_boxin.jpg image
html
local
a
href class title
/wiki/File:Edificio_Fuller_(Flatiron)_en_2010_desde_el_Empire_State_crop_boxin.jpg internal Enlarge
html
local
div
class
magnify
html
local
a
href title
/wiki/Flatiron_Building Flatiron Building
html
local
a
href title
/wiki/Triangular_prism Triangular prism
html
local
div
class
thumbcaption
html
local
div
class style
thumb tright thumbinner width:222px;
html
local
a
href title
/wiki/Rectangle Rectangle
html
local
a
href title
/wiki/Architect Architect
html
local
a
href title
/wiki/Earthquake Earthquake
html
local
a
href
#cite_note-twsMarE34-41
html
local
sup
id class
cite_ref-twsMarE34_41-0 reference
html
local
p
html
local
a
href title
/wiki/New_York_City New York City
html
local
a
href title
/wiki/Broadway_(Manhattan) Broadway (Manhattan)
html
local
a
href title
/wiki/Flatiron_Building Flatiron Building
html
local
a
href
#cite_note-twsMarE32-42
html
local
sup
id class
cite_ref-twsMarE32_42-0 reference
html
local
a
href title
/wiki/Norway Norway
html
local
a
href
#cite_note-twsMarE35-43
html
local
sup
id class
cite_ref-twsMarE35_43-0 reference
html
local
a
href
#cite_note-twsMarE36-44
html
local
sup
id class
cite_ref-twsMarE36_44-0 reference
html
local
a
href
#cite_note-twsMarE37-45
html
local
sup
id class
cite_ref-twsMarE37_45-0 reference
html
local
a
href
#cite_note-twsMarE41-46
html
local
sup
id class
cite_ref-twsMarE41_46-0 reference
html
local
p
html
local
a
href title
/wiki/Parallelogram Parallelogram
html
local
a
href class title
/wiki/Bricks mw-redirect Bricks
html
local
a
href
#cite_note-twsMarE33-47
html
local
sup
id class
cite_ref-twsMarE33_47-0 reference
html
local
a
href title
/wiki/Tessellation Tessellation
html
local
a
href title
/wiki/Hexagon Hexagon
html
local
a
href title
/wiki/Nature Nature
html
local
a
href title
/wiki/Cantilever Cantilever
html
local
a
href title
/wiki/Space_frame Space frame
html
local
p
html
local
span
class id
mw-headline See_also
html
local
h2
html
local
a
href class title
/wiki/Apollonius%27_theorem mw-redirect Apollonius' theorem
html
local
li
html
local
a
href title
/wiki/Congruence_(geometry) Congruence (geometry)
html
local
li
html
local
a
href class title
/wiki/Desargues%27_theorem mw-redirect Desargues' theorem
html
local
li
html
local
a
href title
/wiki/Dragon%27s_Eye_(symbol) Dragon's Eye (symbol)
html
local
li
html
local
a
href title
/wiki/Fermat_point Fermat point
html
local
li
html
local
a
href title
/wiki/Hadwiger%E2%80%93Finsler_inequality Hadwiger–Finsler inequality
html
local
li
html
local
a
href title
/wiki/Heronian_triangle Heronian triangle
html
local
li
html
local
ul
html
local
td
style
text-align: left; vertical-align: top;
html
local
a
href title
/wiki/Integer_triangle Integer triangle
html
local
li
html
local
a
href title
/wiki/Law_of_cosines Law of cosines
html
local
li
html
local
a
href title
/wiki/Law_of_sines Law of sines
html
local
li
html
local
a
href title
/wiki/Law_of_tangents Law of tangents
html
local
li
html
local
a
href title
/wiki/Lester%27s_theorem Lester's theorem
html
local
li
html
local
a
href title
/wiki/List_of_triangle_inequalities List of triangle inequalities
html
local
li
html
local
a
href title
/wiki/List_of_triangle_topics List of triangle topics
html
local
li
html
local
a
href title
/wiki/Ono%27s_inequality Ono's inequality
html
local
li
html
local
ul
html
local
td
style
text-align: left; vertical-align: top;
html
local
a
href title
/wiki/Pedal_triangle Pedal triangle
html
local
li
html
local
a
href title
/wiki/Pedoe%27s_inequality Pedoe's inequality
html
local
li
html
local
a
href title
/wiki/Pythagorean_theorem Pythagorean theorem
html
local
li
html
local
a
href class title
/wiki/Special_right_triangles mw-redirect Special right triangles
html
local
li
html
local
a
href title
/wiki/Triangle_center Triangle center
html
local
li
html
local
a
href title
/wiki/Triangular_number Triangular number
html
local
li
html
local
a
href title
/wiki/Triangulated_category Triangulated category
html
local
li
html
local
a
href title
/wiki/Triangulation_(topology) Triangulation (topology)
html
local
li
html
local
ul
html
local
td
style
text-align: left; vertical-align: top;
html
local
tr
html
local
tbody
html
local
table
class role style
multicol presentation "border-collapse:
html
local
div
html
local
span
class id
mw-headline Notes
html
local
h2
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-2
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
i
html
local
a
href class title
https://en.wiktionary.org/wiki/Isosceles_triangle extiw wikt:Isosceles triangle
html
local
span
class
reference-text
html
local
li
id
cite_note-2
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-7
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
i
html
local
a
href class title
https://en.wiktionary.org/wiki/convex extiw wikt:convex
html
local
span
class
reference-text
html
local
li
id
cite_note-7
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-8
html
local
b
html
local
span
class
mw-cite-backlink
html
local
span
class
reference-text
html
local
li
id
cite_note-8
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-9
html
local
b
html
local
span
class
mw-cite-backlink
html
local
span
class
reference-text
html
local
li
id
cite_note-9
html
local
ol
class
references
html
local
div
class style
reflist mw-references-wrap list-style-type: decimal;
html
local
span
class id
mw-headline References
html
local
h2
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-1
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href title
/wiki/Eric_W._Weisstein Eric W. Weisstein
html
local
a
rel class href
nofollow external text http://mathworld.wolfram.com/EquilateralTriangle.html
html
local
a
href title
/wiki/MathWorld MathWorld
html
local
i
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Equilateral+Triangle&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FEquilateralTriangle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class id
citation mathworld Reference-Mathworld-Equilateral_Triangle
html
local
span
class
reference-text
html
local
li
id
cite_note-1
html
local
span
class
cite-accessibility-label
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-MWisosceles_3-0
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-MWisosceles_3-1
html
local
span
class
mw-cite-backlink
html
local
a
href title
/wiki/Eric_W._Weisstein Eric W. Weisstein
html
local
a
rel class href
nofollow external text http://mathworld.wolfram.com/IsoscelesTriangle.html
html
local
a
href title
/wiki/MathWorld MathWorld
html
local
i
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Isosceles+Triangle&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FIsoscelesTriangle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class id
citation mathworld Reference-Mathworld-Isosceles_Triangle
html
local
span
class
reference-text
html
local
li
id
cite_note-MWisosceles-3
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-4
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href title
/wiki/Eric_W._Weisstein Eric W. Weisstein
html
local
a
rel class href
nofollow external text http://mathworld.wolfram.com/ScaleneTriangle.html
html
local
a
href title
/wiki/MathWorld MathWorld
html
local
i
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Scalene+triangle&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FScaleneTriangle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class id
citation mathworld Reference-Mathworld-Scalene_triangle
html
local
span
class
reference-text
html
local
li
id
cite_note-4
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-5
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href class title
/wiki/Oxford_Users%27_Guide_to_Mathematics mw-redirect Oxford Users' Guide to Mathematics
html
local
i
html
local
a
href title
/wiki/Oxford_University_Press Oxford University Press
html
local
a
href title
/wiki/International_Standard_Book_Number International Standard Book Number
html
local
a
href title
/wiki/Special:BookSources/978-0-19-850763-5 Special:BookSources/978-0-19-850763-5
html
local
cite
class
citation book
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+Users%27+Guide+to+Mathematics&rft.pages=729&rft.pub=Oxford+University+Press&rft.date=2004&rft.isbn=978-0-19-850763-5&rft.aulast=Zeidler&rft.aufirst=Eberhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-5
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-6
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI32.html
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Euclid%27s+Elements%2C+Book+I%2C+Proposition+32&rft_id=http%3A%2F%2Faleph0.clarku.edu%2F~djoyce%2Fjava%2Felements%2FbookI%2FpropI32.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-6
html
local
span
class
cite-accessibility-label
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-VV_10-0
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-VV_10-1
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-VV-10
html
local
span
class
cite-accessibility-label
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-LH_11-0
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-LH_11-1
html
local
span
class
mw-cite-backlink
html
local
a
href class title
/wiki/Mathematical_Gazette mw-redirect Mathematical Gazette
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-LH-11
html
local
span
class
cite-accessibility-label
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-LawCosSin_12-0
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-LawCosSin_12-1
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://www.clarku.edu/~djoyce/trig/laws.html
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Laws+of+Cosines+and+Sines&rft.pub=Clark+University&rft.au=Prof.+David+E.+Joyce&rft_id=http%3A%2F%2Fwww.clarku.edu%2F~djoyce%2Ftrig%2Flaws.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-LawCosSin-12
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-13
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href title
/wiki/Eric_W._Weisstein Eric W. Weisstein
html
local
a
rel class href
nofollow external text http://mathworld.wolfram.com/LawofTangents.html
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Law+of+Tangents&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FLawofTangents.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-13
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-14
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href title
/wiki/Eric_W._Weisstein Eric W. Weisstein
html
local
a
rel class href
nofollow external text http://mathworld.wolfram.com/TriangleArea.html
html
local
a
href title
/wiki/MathWorld MathWorld
html
local
i
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Triangle+area&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FTriangleArea.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class id
citation mathworld Reference-Mathworld-Triangle_area
html
local
span
class
reference-text
html
local
li
id
cite_note-14
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-15
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href title
/wiki/Eric_W._Weisstein Eric W. Weisstein
html
local
a
rel class href
nofollow external text http://mathworld.wolfram.com/Triangle.html
html
local
a
href title
/wiki/MathWorld MathWorld
html
local
i
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Triangle&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FTriangle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class id
citation mathworld Reference-Mathworld-Triangle
html
local
span
class
reference-text
html
local
li
id
cite_note-15
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-16
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://www.maa.org/pubs/Calc_articles/ma063.pdf
html
local
span
style
font-size:85%;
html
local
i
html
local
b
html
local
a
href title
/wiki/Digital_object_identifier Digital object identifier
html
local
a
rel class href
nofollow external text //doi.org/10.2307%2F2686282
html
local
cite
class
citation journal
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+College+Mathematics+Journal&rft.atitle=The+Surveyor%27s+Area+Formula&rft.volume=17&rft.issue=4&rft.pages=326-337&rft.date=1986&rft_id=info%3Adoi%2F10.2307%2F2686282&rft.au=Bart+Braden&rft_id=http%3A%2F%2Fwww.maa.org%2Fpubs%2FCalc_articles%2Fma063.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-16
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-17
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-17
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-18
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-18
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-19
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-19
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-Kiss_20-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
a
rel class href
nofollow external text http://forumgeom.fau.edu/FG2016volume16/FG201634.pdf
html
local
span
class
reference-text
html
local
li
id
cite_note-Kiss-20
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-21
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text https://web.archive.org/web/20130620011530/http://www.artofproblemsolving.com/Wiki/index.php/Circumradius
html
local
i
html
local
a
rel class href
nofollow external text http://www.artofproblemsolving.com/Wiki/index.php/Circumradius
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=AoPSWiki&rft.atitle=Circumradius&rft_id=http%3A%2F%2Fwww.artofproblemsolving.com%2FWiki%2Findex.php%2FCircumradius&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-21
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-22
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-22
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-23
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href class title
/wiki/Mathematical_Gazette mw-redirect Mathematical Gazette
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-23
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-24
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-24
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-25
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-25
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-26
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-26
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-27
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-27
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-Dunn_28-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href class title
/wiki/Mathematical_Gazette mw-redirect Mathematical Gazette
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-Dunn-28
html
local
span
class
cite-accessibility-label
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-0
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-1
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-2
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-3
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-4
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-5
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-6
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Altshiller-Court_29-7
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-Altshiller-Court-29
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-30
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
a
rel class href
nofollow external text http://forumgeom.fau.edu/FG2004volume4/FG200425.pdf
html
local
span
class
reference-text
html
local
li
id
cite_note-30
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-31
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-31
html
local
span
class
cite-accessibility-label
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Johnson_32-0
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Johnson_32-1
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-Johnson_32-2
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-Johnson-32
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-33
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href
#CITEREFAltshiller-Court1925
html
local
span
class
reference-text
html
local
li
id
cite_note-33
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-34
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3338&pf=1
html
local
a
href title
/wiki/American_Mathematical_Monthly American Mathematical Monthly
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-34
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-35
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-35
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-36
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://mathworld.wolfram.com/TriangleCircumscribing.html
html
local
i
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+Math+World&rft.atitle=Triangle+Circumscribing&rft.aulast=Weisstein&rft.aufirst=Eric+W&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FTriangleCircumscribing.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-36
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-37
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
href title
/wiki/Mathematics_Magazine Mathematics Magazine
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-37
html
local
span
class
cite-accessibility-label
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-OS_38-0
html
local
b
html
local
i
html
local
sup
html
local
a
href
#cite_ref-OS_38-1
html
local
span
class
mw-cite-backlink
html
local
i
html
local
a
rel class href
nofollow external text http://forumgeom.fau.edu/FG2013volume13/FG201311index.html
html
local
span
class
reference-text
html
local
li
id
cite_note-OS-38
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-39
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://math.stackexchange.com/a/269544/29780
html
local
i
html
local
cite
class
citation web
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+Stack+Exchange&rft.atitle=Is+the+area+of+intersection+of+convex+polygons+always+convex%3F&rft.aulast=-&rft.aufirst=Christos&rft_id=http%3A%2F%2Fmath.stackexchange.com%2Fa%2F269544%2F29780&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-39
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-40
html
local
b
html
local
span
class
mw-cite-backlink
html
local
i
html
local
span
class
reference-text
html
local
li
id
cite_note-40
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-twsMarE34_41-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://articles.latimes.com/1989-11-10/business/fi-1169_1_stories-tokyo-design
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
q
html
local
cite
class
citation news
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Tokyo+Designers+Envision+500-Story+Tower&rft.date=1989-11-10&rft.au=Associated+Press&rft_id=http%3A%2F%2Farticles.latimes.com%2F1989-11-10%2Fbusiness%2Ffi-1169_1_stories-tokyo-design&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-twsMarE34-41
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-twsMarE32_42-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text https://www.nytimes.com/2010/05/26/realestate/commercial/26flatiron.html
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
q
html
local
cite
class
citation news
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Quirky+Building+That+Has+Charmed+Its+Tenants&rft.date=2010-05-26&rft.au=Stapinski%2C+Helene&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2010%2F05%2F26%2Frealestate%2Fcommercial%2F26flatiron.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-twsMarE32-42
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-twsMarE35_43-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://www.architectureweek.com/2010/1215/design_2-1.html
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
q
html
local
cite
class
citation news
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Triangle+House+in+Norway&rft.date=2009&rft.aulast=Jodidio&rft.aufirst=Philip&rft_id=http%3A%2F%2Fwww.architectureweek.com%2F2010%2F1215%2Fdesign_2-1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-twsMarE35-43
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-twsMarE36_44-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://archrecord.construction.com/projects/portfolio/archives/0907chapel-1.asp
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
q
html
local
cite
class
citation news
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Chapel+of+the+Deaconesses+of+Reuilly&rft.date=2009-07&rft.aulast=Metz&rft.aufirst=Tracy&rft_id=http%3A%2F%2Farchrecord.construction.com%2Fprojects%2Fportfolio%2Farchives%2F0907chapel-1.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-twsMarE36-44
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-twsMarE37_45-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://archrecord.construction.com/features/digital/archives/0508dignews-1.asp
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
q
html
local
cite
class
citation news
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Tech+Briefs%3A+Seismic+framing+technology+and+smart+siting+aid+a+California+community+college&rft.date=2011-03-05&rft.au=Deborah+Snoonian%2C+P.E.&rft_id=http%3A%2F%2Farchrecord.construction.com%2Ffeatures%2Fdigital%2Farchives%2F0508dignews-1.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-twsMarE37-45
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-twsMarE41_46-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel class href
nofollow external text http://archrecord.construction.com/projects/bts/archives/civic/06_prairieridge/default.asp
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
q
html
local
cite
class
citation news
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Prairie+Ridge+Ecostation+for+Wildlife+and+Learning&rft.date=2006-11&rft.au=Sarah+Amelar&rft_id=http%3A%2F%2Farchrecord.construction.com%2Fprojects%2Fbts%2Farchives%2Fcivic%2F06_prairieridge%2Fdefault.asp&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-twsMarE41-46
html
local
span
class
cite-accessibility-label
html
local
a
href
#cite_ref-twsMarE33_47-0
html
local
b
html
local
span
class
mw-cite-backlink
html
local
a
rel href class
nofollow http://www.boston.com/bostonglobe/ideas/articles/2011/03/13/building_a_better_brick/ external text
html
local
i
html
local
span
class
nowrap
html
local
span
class
reference-accessdate
html
local
q
html
local
cite
class
citation news
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Building+a+better+brick&rft.date=2011-03-13&rft.au=Joshua+Rothman&rft_id=http%3A%2F%2Fwww.boston.com%2Fbostonglobe%2Fideas%2Farticles%2F2011%2F03%2F13%2Fbuilding_a_better_brick%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
span
class
reference-text
html
local
li
id
cite_note-twsMarE33-47
html
local
ol
class
references
html
local
div
class style
reflist columns references-column-width -moz-column-width: 35em; -webkit-column-width: 35em; column-width: 35em; list-style-type: decimal;
html
local
span
class id
mw-headline External_links
html
local
h2
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 30 40 noviewer //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x 1024 1376
html
local
td
class
mbox-image
html
local
a
href class title
https://commons.wikimedia.org/wiki/Category:Triangles extiw commons:Category:Triangles
html
local
b
html
local
i
html
local
td
class
mbox-text plainlist
html
local
tr
html
local
tbody
html
local
table
role class style
presentation mbox-small plainlinks sistersitebox background-color:#f9f9f9;border:1px solid #aaa;color:#000
html
local
img
alt src width height class srcset data-file-width data-file-height
//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png 40 40 noviewer //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x 512 512
html
local
td
class
mbox-image
html
local
a
href class title
https://en.wiktionary.org/wiki/Special:Search/triangle extiw wiktionary:Special:Search/triangle
html
local
b
html
local
i
html
local
td
class
mbox-text plainlist
html
local
tr
html
local
tbody
html
local
table
role class style
presentation mbox-small plainlinks sistersitebox background-color:#f9f9f9;border:1px solid #aaa;color:#000
html
local
a
rel class href
nofollow external text https://www.encyclopediaofmath.org/index.php?title=Triangle&oldid=18404
html
local
a
href title
/wiki/Michiel_Hazewinkel Michiel Hazewinkel
html
local
a
href title
/wiki/Encyclopedia_of_Mathematics Encyclopedia of Mathematics
html
local
i
html
local
a
href title
/wiki/International_Standard_Book_Number International Standard Book Number
html
local
a
href title
/wiki/Special:BookSources/978-1-55608-010-4 Special:BookSources/978-1-55608-010-4
html
local
cite
id class
CITEREFIvanov2001 citation
html
local
span
style
display:none;
html
local
span
title class
ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Triangle&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=Springer+Science%2BBusiness+Media+B.V.+%2F+Kluwer+Academic+Publishers&rft.date=2001&rft.isbn=978-1-55608-010-4&rft.aulast=Ivanov&rft.aufirst=A.B.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTriangle%26oldid%3D18404&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle Z3988
html
local
li
html
local
a
rel class href
nofollow external text https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
html
local
li
html
local
ul
html
local
a
id href
collapseButton0 #
html
local
span
class
collapseButton
html
local
abbr
title style
View this template ;;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none;
html
local
a
href title
/wiki/Template:Polygons Template:Polygons
html
local
li
class
nv-view
html
local
abbr
title style
Discuss this template ;;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none;
html
local
a
href title
/wiki/Template_talk:Polygons Template talk:Polygons
html
local
li
class
nv-talk
html
local
abbr
title style
Edit this template ;;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none;
html
local
a
class href
external text //en.wikipedia.org/w/index.php?title=Template:Polygons&action=edit
html
local
li
class
nv-edit
html
local
ul
html
local
div
class
plainlinks hlist navbar mini
html
local
a
href title
/wiki/Polygon Polygon
html
local
div
id style
Polygons font-size:114%;margin:0 4em
html
local
th
scope class colspan
col navbox-title 2
html
local
tr
html
local
a
href title
/wiki/Regular_polygon Regular polygon
html
local
li
html
local
a
href title
/wiki/List_of_polygons List of polygons
html
local
li
html
local
ul
html
local
div
class
hlist hlist-separated
html
local
td
class colspan
navbox-abovebelow 2
html
local
tr
html
local
th
scope class style
row navbox-group width:1%
html
local
a
href title
/wiki/Monogon Monogon
html
local
li
html
local
a
href title
/wiki/Digon Digon
html
local
li
html
local
a
class
mw-selflink selflink
html
local
a
href title
/wiki/Equilateral_triangle Equilateral triangle
html
local
li
html
local
a
href title
/wiki/Isosceles_triangle Isosceles triangle
html
local
li
html
local
ul
html
local
li
html
local
a
href title
/wiki/Quadrilateral Quadrilateral
html
local
a
href title
/wiki/Square Square
html
local
li
html
local
a
href title
/wiki/Rectangle Rectangle
html
local
li
html
local
a
href title
/wiki/Rhombus Rhombus
html
local
li
html
local
a
href title
/wiki/Parallelogram Parallelogram
html
local
li
html
local
a
href title
/wiki/Trapezoid Trapezoid
html
local
li
html
local
a
href title
/wiki/Kite_(geometry) Kite (geometry)
html
local
li
html
local
ul
html
local
li
html
local
a
href title
/wiki/Pentagon Pentagon
html
local
li
html
local
a
href title
/wiki/Hexagon Hexagon
html
local
li
html
local
a
href title
/wiki/Heptagon Heptagon
html
local
li
html
local
a
href title
/wiki/Octagon Octagon
html
local
li
html
local
a
href title
/wiki/Nonagon Nonagon
html
local
li
html
local
a
href title
/wiki/Decagon Decagon
html
local
li
html
local
ul
html
local
div
style
padding:0em 0.25em
html
local
td
class style
navbox-list navbox-odd hlist text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px
html
local
tr
html
local
th
scope class style
row navbox-group width:1%
html
local
a
href title
/wiki/Hendecagon Hendecagon
html
local
li
html
local
a
href title
/wiki/Dodecagon Dodecagon
html
local
li
html
local
a
href title
/wiki/Tridecagon Tridecagon
html
local
li
html
local
a
href title
/wiki/Tetradecagon Tetradecagon
html
local
li
html
local
a
href title
/wiki/Pentadecagon Pentadecagon
html
local
li
html
local
a
href title
/wiki/Hexadecagon Hexadecagon
html
local
li
html
local
a
href title
/wiki/Heptadecagon Heptadecagon
html
local
li
html
local
a
href title
/wiki/Octadecagon Octadecagon
html
local
li
html
local
a
href title
/wiki/Enneadecagon Enneadecagon
html
local
li
html
local
a
href title
/wiki/Icosagon Icosagon
html
local
li
html
local
ul
html
local
div
style
padding:0em 0.25em
html
local
td
class style
navbox-list navbox-even hlist text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px
html
local
tr
html
local
br
html
local
th
scope class style
row navbox-group width:1%
html
local
a
href title
/wiki/Icosidigon Icosidigon
html
local
li
html
local
a
href title
/wiki/Icositetragon Icositetragon
html
local
li
html
local
a
href title
/wiki/Icosihexagon Icosihexagon
html
local
li
html
local
a
href title
/wiki/Icosioctagon Icosioctagon
html
local
li
html
local
a
href title
/wiki/Triacontagon Triacontagon
html
local
li
html
local
a
href title
/wiki/Triacontadigon Triacontadigon
html
local
li
html
local
a
href title
/wiki/Triacontatetragon Triacontatetragon
html
local
li
html
local
a
href title
/wiki/Tetracontagon Tetracontagon
html
local
li
html
local
a
href title
/wiki/Tetracontadigon Tetracontadigon
html
local
li
html
local
a
href title
/wiki/Tetracontaoctagon Tetracontaoctagon
html
local
li
html
local
a
href title
/wiki/Pentacontagon Pentacontagon
html
local
li
html
local
a
href title
/wiki/Hexacontagon Hexacontagon
html
local
li
html
local
a
href title
/wiki/Hexacontatetragon Hexacontatetragon
html
local
li
html
local
a
href title
/wiki/Heptacontagon Heptacontagon
html
local
li
html
local
a
href title
/wiki/Octacontagon Octacontagon
html
local
li
html
local
a
href title
/wiki/Enneacontagon Enneacontagon
html
local
li
html
local
a
href title
/wiki/Enneacontahexagon Enneacontahexagon
html
local
li
html
local
a
href title
/wiki/Hectogon Hectogon
html
local
li
html
local
ul
html
local
div
style
padding:0em 0.25em
html
local
td
class style
navbox-list navbox-odd hlist text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px
html
local
tr
html
local
th
scope class style
row navbox-group width:1%
html
local
a
href title
/wiki/120-gon 120-gon
html
local
li
html
local
a
href title
/wiki/257-gon 257-gon
html
local
li
html
local
a
href title
/wiki/360-gon 360-gon
html
local
li
html
local
a
href title
/wiki/Chiliagon Chiliagon
html
local
li
html
local
a
href title
/wiki/Myriagon Myriagon
html
local
li
html
local
a
href title
/wiki/65537-gon 65537-gon
html
local
li
html
local
a
href title
/wiki/Megagon Megagon
html
local
li
html
local
a
href title
/wiki/Apeirogon Apeirogon
html
local
li
html
local
ul
html
local
div
style
padding:0em 0.25em
html
local
td
class style
navbox-list navbox-even hlist text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px
html
local
tr
html
local
a
href title
/wiki/Star_polygon Star polygon
html
local
br
html
local
th
scope class style
row navbox-group width:1%
html
local
a
href title
/wiki/Pentagram Pentagram
html
local
li
html
local
a
href title
/wiki/Hexagram Hexagram
html
local
li
html
local
a
href title
/wiki/Heptagram Heptagram
html
local
li
html
local
a
href title
/wiki/Octagram Octagram
html
local
li
html
local
a
href title
/wiki/Enneagram_(geometry) Enneagram (geometry)
html
local
li
html
local
a
href title
/wiki/Decagram_(geometry) Decagram (geometry)
html
local
li
html
local
a
href title
/wiki/Hendecagram Hendecagram
html
local
li
html
local
a
href title
/wiki/Dodecagram Dodecagram
html
local
li
html
local
ul
html
local
div
style
padding:0em 0.25em
html
local
td
class style
navbox-list navbox-odd hlist text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px
html
local
tr
html
local
tbody
html
local
table
class style id
nowraplinks collapsible autocollapse navbox-inner border-spacing:0;background:transparent;color:inherit collapsibleTable0
html
local
div
role class aria-labelledby style
navigation navbox Polygons padding:3px
html
local
div
class
mw-parser-output
html
local
img
src alt title width height style
//en.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1 1 1 border: none; position: absolute;
html
local
noscript
html
local
div
id lang dir class
mw-content-text en ltr mw-content-ltr
html
local
a
dir href
ltr https://en.wikipedia.org/w/index.php?title=Triangle&oldid=835178499
html
local
div
class
printfooter
html
local
a
href title
/wiki/Help:Category Help:Category
html
local
a
href title
/wiki/Category:Triangles Category:Triangles
html
local
li
html
local
a
href title
/wiki/Category:Elementary_shapes Category:Elementary shapes
html
local
li
html
local
a
href title
/wiki/Category:Triangle_geometry Category:Triangle geometry
html
local
li
html
local
ul
html
local
div
id class
mw-normal-catlinks mw-normal-catlinks
html
local
a
href title
/wiki/Category:Wikipedia_pages_semi-protected_against_vandalism Category:Wikipedia pages semi-protected against vandalism
html
local
li
html
local
a
href title
/wiki/Category:Wikipedia_indefinitely_move-protected_pages Category:Wikipedia indefinitely move-protected pages
html
local
li
html
local
a
href title
/wiki/Category:Use_dmy_dates_from_August_2014 Category:Use dmy dates from August 2014
html
local
li
html
local
ul
html
local
div
id class
mw-hidden-catlinks mw-hidden-catlinks mw-hidden-cats-hidden
html
local
div
id class data-mw
catlinks catlinks interface
html
local
div
class
visualClear
html
local
div
id class
bodyContent mw-body-content
html
local
div
id class role
content mw-body main
html
local
h2
html
local
h3
id
p-personal-label
html
local
li
id
pt-anonuserpage
html
local
a
href title accesskey
/wiki/Special:MyTalk Discussion about edits from this IP address [ctrl-alt-n] n
html
local
li
id
pt-anontalk
html
local
a
href title accesskey
/wiki/Special:MyContributions A list of edits made from this IP address [ctrl-alt-y] y
html
local
li
id
pt-anoncontribs
html
local
a
href title
/w/index.php?title=Special:CreateAccount&returnto=Triangle You are encouraged to create an account and log in; however, it is not mandatory
html
local
li
id
pt-createaccount
html
local
a
href title accesskey
/w/index.php?title=Special:UserLogin&returnto=Triangle You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o] o
html
local
li
id
pt-login
html
local
ul
html
local
div
id role class aria-labelledby
p-personal navigation p-personal-label
html
local
h3
id
p-namespaces-label
html
local
a
href title accesskey
/wiki/Triangle View the content page [ctrl-alt-c] c
html
local
span
html
local
li
id class
ca-nstab-main selected
html
local
a
href rel title accesskey
/wiki/Talk:Triangle discussion Discussion about the content page [ctrl-alt-t] t
html
local
span
html
local
li
id
ca-talk
html
local
ul
html
local
div
id role class aria-labelledby
p-namespaces navigation vectorTabs p-namespaces-label
html
local
input
type class aria-labelledby
checkbox vectorMenuCheckbox p-variants-label
html
local
span
html
local
h3
id
p-variants-label
html
local
ul
html
local
div
class
menu
html
local
div
id role class aria-labelledby
p-variants navigation vectorMenu emptyPortlet p-variants-label
html
local
div
id
left-navigation
html
local
h3
id
p-views-label
html
local
a
href
/wiki/Triangle
html
local
span
html
local
li
id class
ca-view collapsible selected
html
local
a
href title accesskey
/w/index.php?title=Triangle&action=edit This page is protected.
You can view its source [ctrl-alt-e] e
html
local
span
html
local
li
id class
ca-viewsource collapsible
html
local
a
href title accesskey
/w/index.php?title=Triangle&action=history Past revisions of this page [ctrl-alt-h] h
html
local
span
html
local
li
id class
ca-history collapsible
html
local
ul
html
local
div
id role class aria-labelledby
p-views navigation vectorTabs p-views-label
html
local
input
type class aria-labelledby
checkbox vectorMenuCheckbox p-cactions-label
html
local
span
html
local
h3
id
p-cactions-label
html
local
ul
html
local
div
class
menu
html
local
div
id role class aria-labelledby
p-cactions navigation vectorMenu emptyPortlet p-cactions-label
html
local
label
for
searchInput
html
local
h3
html
local
div
id role
p-search search
html
local
div
id
right-navigation
html
local
div
id
mw-head
html
local
a
class href title
mw-wiki-logo /wiki/Main_Page Visit the main page
html
local
div
id role
p-logo banner
html
local
h3
id
p-navigation-label
html
local
a
href title accesskey
/wiki/Main_Page Visit the main page [ctrl-alt-z] z
html
local
li
id
n-mainpage-description
html
local
a
href title
/wiki/Portal:Contents Guides to browsing Wikipedia
html
local
li
id
n-contents
html
local
a
href title
/wiki/Portal:Featured_content Featured content – the best of Wikipedia
html
local
li
id
n-featuredcontent
html
local
a
href title
/wiki/Portal:Current_events Find background information on current events
html
local
li
id
n-currentevents
html
local
a
href title accesskey
/wiki/Special:Random Load a random article [ctrl-alt-x] x
html
local
li
id
n-randompage
html
local
a
href title
https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en Support us
html
local
li
id
n-sitesupport
html
local
a
href title
//shop.wikimedia.org Visit the Wikipedia store
html
local
li
id
n-shoplink
html
local
ul
html
local
div
class
body
html
local
div
class role id aria-labelledby
portal navigation p-navigation p-navigation-label
html
local
h3
id
p-interaction-label
html
local
a
href title
/wiki/Help:Contents Guidance on how to use and edit Wikipedia
html
local
li
id
n-help
html
local
a
href title
/wiki/Wikipedia:About Find out about Wikipedia
html
local
li
id
n-aboutsite
html
local
a
href title
/wiki/Wikipedia:Community_portal About the project, what you can do, where to find things
html
local
li
id
n-portal
html
local
a
href title accesskey
/wiki/Special:RecentChanges A list of recent changes in the wiki [ctrl-alt-r] r
html
local
li
id
n-recentchanges
html
local
a
href title
//en.wikipedia.org/wiki/Wikipedia:Contact_us How to contact Wikipedia
html
local
li
id
n-contactpage
html
local
ul
html
local
div
class
body
html
local
div
class role id aria-labelledby
portal navigation p-interaction p-interaction-label
html
local
h3
id
p-tb-label
html
local
a
href title accesskey
/wiki/Special:WhatLinksHere/Triangle List of all English Wikipedia pages containing links to this page [ctrl-alt-j] j
html
local
li
id
t-whatlinkshere
html
local
a
href rel title accesskey
/wiki/Special:RecentChangesLinked/Triangle nofollow Recent changes in pages linked from this page [ctrl-alt-k] k
html
local
li
id
t-recentchangeslinked
html
local
a
href title accesskey
/wiki/Wikipedia:File_Upload_Wizard Upload files [ctrl-alt-u] u
html
local
li
id
t-upload
html
local
a
href title accesskey
/wiki/Special:SpecialPages A list of all special pages [ctrl-alt-q] q
html
local
li
id
t-specialpages
html
local
a
href title
/w/index.php?title=Triangle&oldid=835178499 Permanent link to this revision of the page
html
local
li
id
t-permalink
html
local
a
href title
/w/index.php?title=Triangle&action=info More information about this page
html
local
li
id
t-info
html
local
a
href title accesskey
https://www.wikidata.org/wiki/Special:EntityPage/Q19821 Link to connected data repository item [ctrl-alt-g] g
html
local
li
id
t-wikibase
html
local
a
href title
/w/index.php?title=Special:CiteThisPage&page=Triangle&id=835178499 Information on how to cite this page
html
local
li
id
t-cite
html
local
ul
html
local
div
class
body
html
local
div
class role id aria-labelledby
portal navigation p-tb p-tb-label
html
local
h3
id
p-coll-print_export-label
html
local
a
href
/w/index.php?title=Special:Book&bookcmd=book_creator&referer=Triangle
html
local
li
id
coll-create_a_book
html
local
a
href
/w/index.php?title=Special:ElectronPdf&page=Triangle&action=show-download-screen
html
local
li
id
coll-download-as-rdf2latex
html
local
a
href title accesskey
/w/index.php?title=Triangle&printable=yes Printable version of this page [ctrl-alt-p] p
html
local
li
id
t-print
html
local
ul
html
local
div
class
body
html
local
div
class role id aria-labelledby
portal navigation p-coll-print_export p-coll-print_export-label
html
local
h3
id
p-wikibase-otherprojects-label
html
local
a
href hreflang
https://commons.wikimedia.org/wiki/Category:Triangles en
html
local
li
class
wb-otherproject-link wb-otherproject-commons
html
local
a
href hreflang
https://en.wikibooks.org/wiki/A_Guide_to_the_GRE/Triangles en
html
local
li
class
wb-otherproject-link wb-otherproject-wikibooks
html
local
ul
html
local
div
class
body
html
local
div
class role id aria-labelledby
portal navigation p-wikibase-otherprojects p-wikibase-otherprojects-label
html
local
button
class title
uls-settings-trigger Language settings
html
local
h3
id
p-lang-label
html
local
a
href title lang hreflang class
https://kbd.wikipedia.org/wiki/%D0%A9%D1%8D%D0%BD%D1%8B%D0%B9 Щэный – Kabardian kbd kbd interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-kbd display: none;
html
local
a
href title lang hreflang class
https://af.wikipedia.org/wiki/Driehoek Driehoek – Afrikaans af af interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-af display: none;
html
local
a
href title lang hreflang class
https://am.wikipedia.org/wiki/%E1%88%B6%E1%88%B5%E1%89%B5_%E1%88%9B%E1%8A%A5%E1%8B%98%E1%8A%95 ሶስት ማእዘን – Amharic am am interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-am display: none;
html
local
a
href title lang hreflang class
https://ar.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB مثلث – Arabic ar ar interlanguage-link-target
html
local
li
class
interlanguage-link interwiki-ar
html
local
a
href title lang hreflang class
https://an.wikipedia.org/wiki/Trianglo Trianglo – Aragonese an an interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-an display: none;
html
local
a
href title lang hreflang class
https://arc.wikipedia.org/wiki/%DC%A1%DC%AC%DC%A0%DC%AC%DC%90 ܡܬܠܬܐ – Aramaic arc arc interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-arc display: none;
html
local
a
href title lang hreflang class
https://as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A7%B0%E0%A6%BF%E0%A6%AD%E0%A7%81%E0%A6%9C ত্ৰিভুজ – Assamese as as interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-as display: none;
html
local
a
href title lang hreflang class
https://ast.wikipedia.org/wiki/Tri%C3%A1ngulu Triángulu – Asturian ast ast interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ast display: none;
html
local
a
href title lang hreflang class
https://ay.wikipedia.org/wiki/Mujina Mujina – Aymara ay ay interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ay display: none;
html
local
a
href title lang hreflang class
https://az.wikipedia.org/wiki/%C3%9C%C3%A7bucaq Üçbucaq – Azerbaijani az az interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-az display: none;
html
local
a
href title lang hreflang class
https://azb.wikipedia.org/wiki/%D8%A7%D9%88%DA%86_%DA%AF%D9%86 اوچ گن – South Azerbaijani azb azb interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-azb display: none;
html
local
a
href title lang hreflang class
https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%AD%E0%A7%81%E0%A6%9C ত্রিভুজ – Bangla bn bn interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-bn display: none;
html
local
a
href title lang hreflang class
https://zh-min-nan.wikipedia.org/wiki/Sa%E2%81%BF-kak-h%C3%AAng Saⁿ-kak-hêng – Chinese (Min Nan) zh-min-nan zh-min-nan interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-zh-min-nan display: none;
html
local
a
href title lang hreflang class
https://ba.wikipedia.org/wiki/%D3%A8%D1%81%D0%BC%D3%A9%D0%B9%D3%A9%D1%88 Өсмөйөш – Bashkir ba ba interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ba display: none;
html
local
a
href title lang hreflang class
https://be.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%85%D0%B2%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D1%96%D0%BA Трохвугольнік – Belarusian be be interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-be display: none;
html
local
a
href title lang hreflang class
https://be-x-old.wikipedia.org/wiki/%D0%A2%D1%80%D1%8B%D0%BA%D1%83%D1%82%D0%BD%D1%96%D0%BA Трыкутнік – Belarusian (Taraškievica orthography) be-x-old be-x-old interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-be-x-old display: none;
html
local
a
href title lang hreflang class
https://bh.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C त्रिभुज – Bhojpuri bh bh interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-bh display: none;
html
local
a
href title lang hreflang class
https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%B8%D0%BA Триъгълник – Bulgarian bg bg interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-bg display: none;
html
local
a
href title lang hreflang class
https://bar.wikipedia.org/wiki/Dreieck Dreieck – Bavarian bar bar interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-bar display: none;
html
local
a
href title lang hreflang class
https://bs.wikipedia.org/wiki/Trougao Trougao – Bosnian bs bs interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-bs display: none;
html
local
a
href title lang hreflang class
https://br.wikipedia.org/wiki/Tric%27horn Tric'horn – Breton br br interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-br display: none;
html
local
a
href title lang hreflang class
https://ca.wikipedia.org/wiki/Triangle Triangle – Catalan ca ca interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ca display: none;
html
local
a
href title lang hreflang class
https://cv.wikipedia.org/wiki/%D0%92%D0%B8%C3%A7_%D0%BA%C4%95%D1%82%D0%B5%D1%81%D0%BB%C4%95%D1%85 Виç кĕтеслĕх – Chuvash cv cv interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-cv display: none;
html
local
a
href title lang hreflang class
https://cs.wikipedia.org/wiki/Troj%C3%BAheln%C3%ADk Trojúhelník – Czech cs cs interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-cs display: none;
html
local
a
href title lang hreflang class
https://sn.wikipedia.org/wiki/Gonyonhatu Gonyonhatu – Shona sn sn interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sn display: none;
html
local
a
href title lang hreflang class
https://co.wikipedia.org/wiki/Triangulu Triangulu – Corsican co co interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-co display: none;
html
local
a
href title lang hreflang class
https://cy.wikipedia.org/wiki/Triongl Triongl – Welsh cy cy interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-cy display: none;
html
local
a
href title lang hreflang class
https://da.wikipedia.org/wiki/Trekant Trekant – Danish da da interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-da display: none;
html
local
a
href title lang hreflang class
https://se.wikipedia.org/wiki/Golmma%C4%8Diegat Golmmačiegat – Northern Sami se se interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-se display: none;
html
local
a
href title lang hreflang class
https://de.wikipedia.org/wiki/Dreieck Dreieck – German de de interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-de display: none;
html
local
a
href title lang hreflang class
https://dsb.wikipedia.org/wiki/T%C5%9Biro%C5%BEk Tśirožk – Lower Sorbian dsb dsb interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-dsb display: none;
html
local
a
href title lang hreflang class
https://et.wikipedia.org/wiki/Kolmnurk Kolmnurk – Estonian et et interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-et display: none;
html
local
a
href title lang hreflang class
https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF Τρίγωνο – Greek el el interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-el display: none;
html
local
a
href title lang hreflang class
https://es.wikipedia.org/wiki/Tri%C3%A1ngulo Triángulo – Spanish es es interlanguage-link-target
html
local
li
class
interlanguage-link interwiki-es
html
local
a
href title lang hreflang class
https://eo.wikipedia.org/wiki/Triangulo Triangulo – Esperanto eo eo interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-eo display: none;
html
local
a
href title lang hreflang class
https://eu.wikipedia.org/wiki/Hiruki Hiruki – Basque eu eu interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-eu display: none;
html
local
a
href title lang hreflang class
https://fa.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB مثلث – Persian fa fa interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-fa display: none;
html
local
a
href title lang hreflang class
https://fo.wikipedia.org/wiki/Tr%C3%ADkantur Tríkantur – Faroese fo fo interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-fo display: none;
html
local
a
href title lang hreflang class
https://fr.wikipedia.org/wiki/Triangle Triangle – French fr fr interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-fr display: none;
html
local
a
href title lang hreflang class
https://ga.wikipedia.org/wiki/Triant%C3%A1n Triantán – Irish ga ga interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ga display: none;
html
local
a
href title lang hreflang class
https://gv.wikipedia.org/wiki/Troorane Troorane – Manx gv gv interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-gv display: none;
html
local
a
href title lang hreflang class
https://gl.wikipedia.org/wiki/Tri%C3%A1ngulo Triángulo – Galician gl gl interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-gl display: none;
html
local
a
href title lang hreflang class
https://gan.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2 三角形 – Gan Chinese gan gan interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-gan display: none;
html
local
a
href title lang hreflang class
https://gu.wikipedia.org/wiki/%E0%AA%A4%E0%AB%8D%E0%AA%B0%E0%AA%BF%E0%AA%95%E0%AB%8B%E0%AA%A3 ત્રિકોણ – Gujarati gu gu interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-gu display: none;
html
local
a
href title lang hreflang class
https://hak.wikipedia.org/wiki/S%C3%A2m-kok-h%C3%ACn Sâm-kok-hìn – Hakka Chinese hak hak interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-hak display: none;
html
local
a
href title lang hreflang class
https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81%ED%98%95 삼각형 – Korean ko ko interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ko display: none;
html
local
a
href title lang hreflang class
https://hy.wikipedia.org/wiki/%D4%B5%D5%BC%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6 Եռանկյուն – Armenian hy hy interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-hy display: none;
html
local
a
href title lang hreflang class
https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C त्रिभुज – Hindi hi hi interlanguage-link-target
html
local
li
class
interlanguage-link interwiki-hi
html
local
a
href title lang hreflang class
https://hsb.wikipedia.org/wiki/T%C5%99ir%C3%B3%C5%BEk Třiróžk – Upper Sorbian hsb hsb interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-hsb display: none;
html
local
a
href title lang hreflang class
https://hr.wikipedia.org/wiki/Trokut Trokut – Croatian hr hr interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-hr display: none;
html
local
a
href title lang hreflang class
https://io.wikipedia.org/wiki/Triangulo Triangulo – Ido io io interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-io display: none;
html
local
a
href title lang hreflang class
https://id.wikipedia.org/wiki/Segitiga Segitiga – Indonesian id id interlanguage-link-target
html
local
li
class
interlanguage-link interwiki-id
html
local
a
href title lang hreflang class
https://is.wikipedia.org/wiki/%C3%9Er%C3%ADhyrningur Þríhyrningur – Icelandic is is interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-is display: none;
html
local
a
href title lang hreflang class
https://it.wikipedia.org/wiki/Triangolo Triangolo – Italian it it interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-it display: none;
html
local
a
href title lang hreflang class
https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%9C%D7%A9 משולש – Hebrew he he interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-he display: none;
html
local
a
href title lang hreflang class
https://jv.wikipedia.org/wiki/Segi_telu Segi telu – Javanese jv jv interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-jv display: none;
html
local
a
href title lang hreflang class
https://kn.wikipedia.org/wiki/%E0%B2%A4%E0%B3%8D%E0%B2%B0%E0%B2%BF%E0%B2%95%E0%B3%8B%E0%B2%A8 ತ್ರಿಕೋನ – Kannada kn kn interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-kn display: none;
html
local
a
href title lang hreflang class
https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%90%E1%83%9B%E1%83%99%E1%83%A3%E1%83%97%E1%83%AE%E1%83%94%E1%83%93%E1%83%98 სამკუთხედი – Georgian ka ka interlanguage-link-target
html
local
li
class title
interlanguage-link interwiki-ka badge-Q17437796 badge-featuredarticle featured article
html
local
a
href title lang hreflang class
https://csb.wikipedia.org/wiki/Trz%C3%ABn%C3%B3rt Trzënórt – Kashubian csb csb interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-csb display: none;
html
local
a
href title lang hreflang class
https://kk.wikipedia.org/wiki/%D2%AE%D1%88%D0%B1%D2%B1%D1%80%D1%8B%D1%88 Үшбұрыш – Kazakh kk kk interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-kk display: none;
html
local
a
href title lang hreflang class
https://sw.wikipedia.org/wiki/Pembetatu Pembetatu – Swahili sw sw interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sw display: none;
html
local
a
href title lang hreflang class
https://ht.wikipedia.org/wiki/Triyang Triyang – Haitian Creole ht ht interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ht display: none;
html
local
a
href title lang hreflang class
https://ku.wikipedia.org/wiki/S%C3%AAgo%C5%9Fe Sêgoşe – Kurdish ku ku interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ku display: none;
html
local
a
href title lang hreflang class
https://lo.wikipedia.org/wiki/%E0%BA%AE%E0%BA%B9%E0%BA%9A%E0%BA%AA%E0%BA%B2%E0%BA%A1%E0%BB%81%E0%BA%88 ຮູບສາມແຈ – Lao lo lo interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-lo display: none;
html
local
a
href title lang hreflang class
https://la.wikipedia.org/wiki/Triangulum Triangulum – Latin la la interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-la display: none;
html
local
a
href title lang hreflang class
https://lv.wikipedia.org/wiki/Trijst%C5%ABris Trijstūris – Latvian lv lv interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-lv display: none;
html
local
a
href title lang hreflang class
https://lt.wikipedia.org/wiki/Trikampis Trikampis – Lithuanian lt lt interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-lt display: none;
html
local
a
href title lang hreflang class
https://li.wikipedia.org/wiki/Driehook Driehook – Limburgish li li interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-li display: none;
html
local
a
href title lang hreflang class
https://ln.wikipedia.org/wiki/Mpanzi-mis%C3%A1to Mpanzi-misáto – Lingala ln ln interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ln display: none;
html
local
a
href title lang hreflang class
https://lmo.wikipedia.org/wiki/Tri%C3%A0ngol Triàngol – Lombard lmo lmo interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-lmo display: none;
html
local
a
href title lang hreflang class
https://hu.wikipedia.org/wiki/H%C3%A1romsz%C3%B6g Háromszög – Hungarian hu hu interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-hu display: none;
html
local
a
href title lang hreflang class
https://mk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B0%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA Триаголник – Macedonian mk mk interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-mk display: none;
html
local
a
href title lang hreflang class
https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8B%E0%B4%A3%E0%B4%82 ത്രികോണം – Malayalam ml ml interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ml display: none;
html
local
a
href title lang hreflang class
https://mr.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3 त्रिकोण – Marathi mr mr interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-mr display: none;
html
local
a
href title lang hreflang class
https://arz.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB مثلث – Egyptian Arabic arz arz interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-arz display: none;
html
local
a
href title lang hreflang class
https://ms.wikipedia.org/wiki/Segi_tiga Segi tiga – Malay ms ms interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ms display: none;
html
local
a
href title lang hreflang class
https://cdo.wikipedia.org/wiki/S%C4%83ng-g%C3%A1e%CC%A4k-h%C3%ACng Săng-gáe̤k-hìng – Min Dong Chinese cdo cdo interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-cdo display: none;
html
local
a
href title lang hreflang class
https://mn.wikipedia.org/wiki/%D0%93%D1%83%D1%80%D0%B2%D0%B0%D0%BB%D0%B6%D0%B8%D0%BD Гурвалжин – Mongolian mn mn interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-mn display: none;
html
local
a
href title lang hreflang class
https://my.wikipedia.org/wiki/%E1%80%90%E1%80%BC%E1%80%AD%E1%80%82%E1%80%B6 တြိဂံ – Burmese my my interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-my display: none;
html
local
a
href title lang hreflang class
https://nl.wikipedia.org/wiki/Driehoek_(meetkunde) Driehoek (meetkunde) – Dutch nl nl interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-nl display: none;
html
local
a
href title lang hreflang class
https://ne.wikipedia.org/wiki/%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AD%E0%A5%81%E0%A4%9C त्रिभुज – Nepali ne ne interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ne display: none;
html
local
a
href title lang hreflang class
https://new.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%95%E0%A5%81%E0%A4%82 स्वकुं – Newari new new interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-new display: none;
html
local
a
href title lang hreflang class
https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2 三角形 – Japanese ja ja interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ja display: none;
html
local
a
href title lang hreflang class
https://no.wikipedia.org/wiki/Trekant Trekant – Norwegian no no interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-no display: none;
html
local
a
href title lang hreflang class
https://nn.wikipedia.org/wiki/Trekant Trekant – Norwegian Nynorsk nn nn interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-nn display: none;
html
local
a
href title lang hreflang class
https://nrm.wikipedia.org/wiki/Trian Trian – Norman nrm nrm interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-nrm display: none;
html
local
a
href title lang hreflang class
https://oc.wikipedia.org/wiki/Triangle Triangle – Occitan oc oc interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-oc display: none;
html
local
a
href title lang hreflang class
https://mhr.wikipedia.org/wiki/%D0%9A%D1%83%D0%BC%D0%BB%D1%83%D0%BA Кумлук – Eastern Mari mhr mhr interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-mhr display: none;
html
local
a
href title lang hreflang class
https://or.wikipedia.org/wiki/%E0%AC%A4%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%AD%E0%AD%81%E0%AC%9C ତ୍ରିଭୁଜ – Odia or or interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-or display: none;
html
local
a
href title lang hreflang class
https://uz.wikipedia.org/wiki/Uchburchak Uchburchak – Uzbek uz uz interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-uz display: none;
html
local
a
href title lang hreflang class
https://pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%BF%E0%A8%95%E0%A9%8B%E0%A8%A8 ਤਿਕੋਨ – Punjabi pa pa interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-pa display: none;
html
local
a
href title lang hreflang class
https://pfl.wikipedia.org/wiki/Draiegg Draiegg – Palatine German pfl pfl interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-pfl display: none;
html
local
a
href title lang hreflang class
https://pnb.wikipedia.org/wiki/%D8%AA%DA%A9%D9%88%D9%86 تکون – Western Punjabi pnb pnb interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-pnb display: none;
html
local
a
href title lang hreflang class
https://ps.wikipedia.org/wiki/%D8%AF%D8%B1%DB%90%DA%85%D9%86%DA%89%DB%8C درېڅنډی – Pashto ps ps interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ps display: none;
html
local
a
href title lang hreflang class
https://jam.wikipedia.org/wiki/Chrayanggl Chrayanggl – Jamaican Creole English jam jam interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-jam display: none;
html
local
a
href title lang hreflang class
https://km.wikipedia.org/wiki/%E1%9E%8F%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%80%E1%9F%84%E1%9E%8E ត្រីកោណ – Khmer km km interlanguage-link-target
html
local
li
class title
interlanguage-link interwiki-km badge-Q17437796 badge-featuredarticle featured article
html
local
a
href title lang hreflang class
https://pl.wikipedia.org/wiki/Tr%C3%B3jk%C4%85t Trójkąt – Polish pl pl interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-pl display: none;
html
local
a
href title lang hreflang class
https://pt.wikipedia.org/wiki/Tri%C3%A2ngulo Triângulo – Portuguese pt pt interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-pt display: none;
html
local
a
href title lang hreflang class
https://ro.wikipedia.org/wiki/Triunghi Triunghi – Romanian ro ro interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ro display: none;
html
local
a
href title lang hreflang class
https://qu.wikipedia.org/wiki/Kimsak%27uchu Kimsak'uchu – Quechua qu qu interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-qu display: none;
html
local
a
href title lang hreflang class
https://rue.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D1%83%D0%B3%D0%BE%D0%BB%D0%BD%D0%B8%D0%BA Триуголник – Rusyn rue rue interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-rue display: none;
html
local
a
href title lang hreflang class
https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA Треугольник – Russian ru ru interlanguage-link-target
html
local
li
class
interlanguage-link interwiki-ru
html
local
a
href title lang hreflang class
https://sco.wikipedia.org/wiki/Triangle Triangle – Scots sco sco interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sco display: none;
html
local
a
href title lang hreflang class
https://sq.wikipedia.org/wiki/Trek%C3%ABnd%C3%ABshi Trekëndëshi – Albanian sq sq interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sq display: none;
html
local
a
href title lang hreflang class
https://scn.wikipedia.org/wiki/Tri%C3%A0nculu Triànculu – Sicilian scn scn interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-scn display: none;
html
local
a
href title lang hreflang class
https://simple.wikipedia.org/wiki/Triangle Triangle – Simple English simple simple interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-simple display: none;
html
local
a
href title lang hreflang class
https://sk.wikipedia.org/wiki/Trojuholn%C3%ADk Trojuholník – Slovak sk sk interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sk display: none;
html
local
a
href title lang hreflang class
https://sl.wikipedia.org/wiki/Trikotnik Trikotnik – Slovenian sl sl interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sl display: none;
html
local
a
href title lang hreflang class
https://szl.wikipedia.org/wiki/Trziek Trziek – Silesian szl szl interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-szl display: none;
html
local
a
href title lang hreflang class
https://so.wikipedia.org/wiki/Saddexagal Saddexagal – Somali so so interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-so display: none;
html
local
a
href title lang hreflang class
https://ckb.wikipedia.org/wiki/%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95 سێگۆشە – Central Kurdish ckb ckb interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ckb display: none;
html
local
a
href title lang hreflang class
https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%83%D0%B3%D0%B0%D0%BE Троугао – Serbian sr sr interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sr display: none;
html
local
a
href title lang hreflang class
https://sh.wikipedia.org/wiki/Trokut Trokut – Serbo-Croatian sh sh interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sh display: none;
html
local
a
href title lang hreflang class
https://su.wikipedia.org/wiki/Juru_tilu Juru tilu – Sundanese su su interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-su display: none;
html
local
a
href title lang hreflang class
https://fi.wikipedia.org/wiki/Kolmio Kolmio – Finnish fi fi interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-fi display: none;
html
local
a
href title lang hreflang class
https://sv.wikipedia.org/wiki/Triangel Triangel – Swedish sv sv interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-sv display: none;
html
local
a
href title lang hreflang class
https://tl.wikipedia.org/wiki/Tatsulok Tatsulok – Tagalog tl tl interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-tl display: none;
html
local
a
href title lang hreflang class
https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%AE%E0%AF%8D முக்கோணம் – Tamil ta ta interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-ta display: none;
html
local
a
href title lang hreflang class
https://tt.wikipedia.org/wiki/%D3%A8%D1%87%D0%BF%D0%BE%D1%87%D0%BC%D0%B0%D0%BA Өчпочмак – Tatar tt tt interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-tt display: none;
html
local
a
href title lang hreflang class
https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%AD%E0%B1%81%E0%B0%9C%E0%B0%82 త్రిభుజం – Telugu te te interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-te display: none;
html
local
a
href title lang hreflang class
https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%AA%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1 รูปสามเหลี่ยม – Thai th th interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-th display: none;
html
local
a
href title lang hreflang class
https://tg.wikipedia.org/wiki/%D0%A1%D0%B5%D0%BA%D1%83%D0%BD%D2%B7%D0%B0 Секунҷа – Tajik tg tg interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-tg display: none;
html
local
a
href title lang hreflang class
https://chr.wikipedia.org/wiki/%E1%8F%A6%E1%8E%A2_%E1%8F%A7%E1%8F%85%E1%8F%8F%E1%8F%AF_%E1%8E%A4%E1%8F%83%E1%8F%B4%E1%8E%A9 ᏦᎢ ᏧᏅᏏᏯ ᎤᏃᏴᎩ – Cherokee chr chr interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-chr display: none;
html
local
a
href title lang hreflang class
https://tr.wikipedia.org/wiki/%C3%9C%C3%A7gen Üçgen – Turkish tr tr interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-tr display: none;
html
local
a
href title lang hreflang class
https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B8%D0%BA Трикутник – Ukrainian uk uk interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-uk display: none;
html
local
a
href title lang hreflang class
https://ur.wikipedia.org/wiki/%D9%85%D8%AB%D9%84%D8%AB مثلث – Urdu ur ur interlanguage-link-target
html
local
li
class
interlanguage-link interwiki-ur
html
local
a
href title lang hreflang class
https://vec.wikipedia.org/wiki/Triango%C5%82o Triangoło – Venetian vec vec interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-vec display: none;
html
local
a
href title lang hreflang class
https://vi.wikipedia.org/wiki/Tam_gi%C3%A1c Tam giác – Vietnamese vi vi interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-vi display: none;
html
local
a
href title lang hreflang class
https://zh-classical.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2 三角形 – Classical Chinese zh-classical zh-classical interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-zh-classical display: none;
html
local
a
href title lang hreflang class
https://vls.wikipedia.org/wiki/Drieoek Drieoek – West Flemish vls vls interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-vls display: none;
html
local
a
href title lang hreflang class
https://war.wikipedia.org/wiki/Trayanggulo Trayanggulo – Waray war war interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-war display: none;
html
local
a
href title lang hreflang class
https://wuu.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2 三角形 – Wu Chinese wuu wuu interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-wuu display: none;
html
local
a
href title lang hreflang class
https://yi.wikipedia.org/wiki/%D7%93%D7%A8%D7%99%D7%99%D7%A2%D7%A7 דרייעק – Yiddish yi yi interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-yi display: none;
html
local
a
href title lang hreflang class
https://yo.wikipedia.org/wiki/An%C3%ADgunm%E1%BA%B9%CC%81ta Anígunmẹ́ta – Yoruba yo yo interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-yo display: none;
html
local
a
href title lang hreflang class
https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2 三角形 – Cantonese zh-yue zh-yue interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-zh-yue display: none;
html
local
a
href title lang hreflang class
https://bat-smg.wikipedia.org/wiki/Tr%C4%97kompis Trėkompis – Samogitian bat-smg bat-smg interlanguage-link-target
html
local
li
class style
interlanguage-link interwiki-bat-smg display: none;
html
local
a
href title lang hreflang class
https://zh.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2 三角形 – Chinese zh zh interlanguage-link-target
html
local
li
class
interlanguage-link interwiki-zh
html
local
button
class title
mw-interlanguage-selector mw-ui-button All languages (initial selection from common choices by you and others)
html
local
ul
html
local
a
href title class
https://www.wikidata.org/wiki/Special:EntityPage/Q19821#sitelinks-wikipedia Edit interlanguage links wbc-editpage
html
local
span
class
wb-langlinks-edit wb-langlinks-link
html
local
div
class
after-portlet after-portlet-lang
html
local
div
class
body
html
local
div
class role id aria-labelledby
portal navigation p-lang p-lang-label
html
local
div
id
mw-panel
html
local
div
id
mw-navigation
html
local
li
id
footer-info-lastmod
html
local
a
rel href
license //en.wikipedia.org/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_License
html
local
a
rel href style
license //creativecommons.org/licenses/by-sa/3.0/ display:none;
html
local
a
href
//wikimediafoundation.org/wiki/Terms_of_Use
html
local
a
href
//wikimediafoundation.org/wiki/Privacy_policy
html
local
a
href
//www.wikimediafoundation.org/
html
local
li
id
footer-info-copyright
html
local
ul
id
footer-info
html
local
a
href class title
https://wikimediafoundation.org/wiki/Privacy_policy extiw wmf:Privacy policy
html
local
li
id
footer-places-privacy
html
local
a
href title
/wiki/Wikipedia:About Wikipedia:About
html
local
li
id
footer-places-about
html
local
a
href title
/wiki/Wikipedia:General_disclaimer Wikipedia:General disclaimer
html
local
li
id
footer-places-disclaimer
html
local
a
href
//en.wikipedia.org/wiki/Wikipedia:Contact_us
html
local
li
id
footer-places-contact
html
local
a
href
https://www.mediawiki.org/wiki/Special:MyLanguage/How_to_contribute
html
local
li
id
footer-places-developers
html
local
a
href
https://wikimediafoundation.org/wiki/Cookie_statement
html
local
li
id
footer-places-cookiestatement
html
local
a
href class
//en.m.wikipedia.org/w/index.php?title=Triangle&mobileaction=toggle_view_mobile noprint stopMobileRedirectToggle
html
local
li
id
footer-places-mobileview
html
local
a
href
#
html
local
li
style
display: none;
html
local
ul
id
footer-places
html
local
img
src srcset width height alt
/static/images/wikimedia-button.png /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x 88 31 Wikimedia Foundation
html
local
a
href
https://wikimediafoundation.org/
html
local
li
id
footer-copyrightico
html
local
img
src alt srcset width height
/static/images/poweredby_mediawiki_88x31.png Powered by MediaWiki /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x 88 31
html
local
a
href
//www.mediawiki.org/
html
local
li
id
footer-poweredbyico
html
local
ul
id class
footer-icons noprint
html
local
div
style
clear: both;
html
local
div
id role
footer contentinfo
html
local
polygon
points
0 8, 10 8, 18 0, 26 8, 1000 8, 1000 1000, 0 1000
html
local
clippath
id
mwe-popups-mask
html
local
polygon
points
0 8, 274 8, 282 0, 290 8, 1000 8, 1000 1000, 0 1000
html
local
clippath
id
mwe-popups-mask-flip
html
local
polygon
points
0 8, 174 8, 182 0, 190 8, 1000 8, 1000 1000, 0 1000
html
local
clippath
id
mwe-popups-landscape-mask
html
local
polygon
points
0 0, 1000 0, 1000 242, 190 242, 182 250, 174 242, 0 242
html
local
clippath
id
mwe-popups-landscape-mask-flip
html
local
defs
html
local
svg
xmlns width height
http://www.w3.org/2000/svg 0 0
html
local
div
id
mwe-popups-svg
html
local
div
class
suggestions-results
html
local
div
class
suggestions-special
html
local
div
class style
suggestions display: none; font-size: 13px;
html
local
body
class
mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Triangle rootpage-Triangle skin-vector action-view
html
local