the intersection of the two sides of an angle is called the angles
vertex
12434
Angle In plane geometry, an angle is the figure formed by two rays, called the "sides" of the angle, sharing a common endpoint, called the "vertex" of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space,
Angle
12434
Angle In plane geometry, an angle is the figure formed by two rays, called the "sides" of the angle, sharing a common endpoint, called the "vertex" of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space,
Angle
15851977
Ex-tangential quadrilateral In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the "extensions" of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its "excircle", its radius the "exradius" and its center the "excenter" ("E" in the figure). The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions
Ex-tangential quadrilateral
18304791
is tangent to all four extended sides. The excenter (center of the tangent circle) lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. Pascal's theorem states that if six arbitrary points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides
Extended side
1349004
the latter case the quadrilateral is a tangential quadrilateral. Each diagonal of a rhombus bisects opposite angles. The excenter of an ex-tangential quadrilateral lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors (supplementary angle bisectors) at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The tangent to a parabola at any point bisects the angle between the line joining the point to the focus and the line from the point and perpendicular to
Bisection
2099140
two equal sides, and for isosceles sets, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the "vertex angle" and the angles that have the base as one of their sides are called the "base angles". The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle
Isosceles triangle
2799683
Internal and external angles In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than 180°, the polygon is called convex. In contrast, an exterior angle (or ) is an angle formed by one side of a
Internal and external angles
2799685
ordinary convex and concave polygons "k" = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution walking around the perimeter. Internal and external angles In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple
Internal and external angles
12462182
of the remote interior angles. This result, which depends upon Euclid's parallel postulate will be referred to as the "High school exterior angle theorem" (HSEAT) to distinguish it from Euclid's exterior angle theorem. Some authors refer to the "High school exterior angle theorem" as the "strong form" of the exterior angle theorem and "Euclid's exterior angle theorem" as the "weak form". A triangle has three corners, called "vertices". The sides of a triangle (line segments) that come together at a vertex form two angles (four angles if you consider the sides of the triangle to be lines instead of line
Exterior angle theorem
10258173
polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians ( 180°, two right angles) ; otherwise, it is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and
Vertex (geometry)
12375368
studied as an affine space, where collinearity and ratios can be studied but not distances; it can be studied as the complex plane using techniques of complex analysis; and so on. Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the "sides" of the angle, sharing a common endpoint, called the "vertex" of the angle. In Euclidean geometry, angles are used to study polygons and
Geometry
18329934
Acute and obtuse triangles An acute triangle is a triangle with all three angles acute (less than 90°). An obtuse triangle is one with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180°, no triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles—triangles that are not right triangles because they have no 90° angle. In all triangles, the centroid—the intersection of the medians, each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of
Acute and obtuse triangles
4282226
end at an ideal vertex represented as an "omega point". Such a pair of sides may also be said to form an angle of zero. A triangle with a zero angle is impossible in Euclidean geometry for straight sides lying on distinct lines. However, such zero angles are possible with tangent circles. A triangle with one ideal vertex is called an omega triangle. Special Triangles with ideal vertices are: A triangle where one vertex is an ideal point, one angle is right: the third angle is the angle of parallelism for the length of the side between the right and
Hyperbolic triangle
1402464
as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex "A", for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex "A", or the excenter of "A". Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system. Polygons with more than three sides do not all have an incircle tangent
Incircle and excircles of a triangle
6663673
example if in a polygon with the side sequence "abcde" and another with the corresponding side sequence "vwxyz" we have vertex angle "A" appearing between sides "a" and "b" then its corresponding vertex angle "V" must appear between sides "v" and "w". Corresponding sides and corresponding angles In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has
Corresponding sides and corresponding angles
15851979
squares, rhombi, and rectangles) can be considered ex-tangential quadrilaterals with infinite exradius since they satisfy the characterizations in the next section, but the excircle cannot be tangent to both pairs of extensions of opposite sides (since they are parallel). Convex quadrilaterals whose side lengths form an arithmetic progression are always ex-tangential as they satisfy the characterization below for adjacent side lengths. A convex quadrilateral is ex-tangential if and only if there are six concurrent angles bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external
Ex-tangential quadrilateral
121155
angle are called complementary. Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. Angles whose sum is a straight angle are supplementary. Supplementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite.
Euclidean geometry
417368
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. 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: Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles
Triangle
7958727
Sum of angles of a triangle In several geometries, a triangle has three "vertices" and three "sides", where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, radians, two right angles, or a half-turn. It was unknown for a long time whether other geometries exist, where this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the
Sum of angles of a triangle
1402475
one of the triangle's sides. The center of an excircle is the intersection of the internal bisector of one angle (at vertex "A", for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex "A", or the excenter of "A". Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system. While A triangle's incenter has trilinear coordinates formula_67 Its excenters (the centers of its excircles) have trilinears
Incircle and excircles of a triangle
12609132
exactly to this border have a vertex touching another side. The concave pentagons are non-stellated pentagons having at least one angle greater than 180°. The first angle which opens wider than 180° is γ, so γ = 180° (border shown in green at right) is a curve which is the border of the regions of concave pentagons and others, called convex. Pentagons which map exactly to this border have at least two consecutive sides appearing as a double length side, which resembles a pentagon degenerated to a quadrilateral. The convex pentagons have all of their five angles smaller than 180°
Equilateral pentagon
8770976
This angle has the property that where formula_3 are the vertex angles formula_4 respectively. There is also a second Brocard point, Q, in triangle "ABC" such that line segments "AQ", "BQ", and "CQ" form equal angles with sides "b", "c", and "a" respectively. In other words, the equations formula_5 apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words, angle formula_6 is the same as formula_7 The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in
Brocard points
12450
of thumb approximations only. The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—"amphicyrtic" (Gr. , on both sides, κυρτός, convex) or "cissoidal" (Gr. κισσός, ivy), biconvex; "xystroidal" or "sistroidal" (Gr. ξυστρίς, a tool for scraping), concavo-convex; "amphicoelic" (Gr. κοίλη, a hollow) or "angulus lunularis", biconcave. The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only
Angle
1348998
the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. Algebraically, the perpendicular bisector of a line segment with endpoints formula_1 and formula_2 is given by the equation which states that the squared distance of a point on the bisector to one endpoint equals the squared distance from that point to the other endpoint. An angle bisector divides the angle into two angles with equal measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The "interior" or "internal bisector" of
Bisection
1349001
The interior angle bisectors of a triangle are concurrent in a point called the incenter of the triangle, as seen in the diagram at right. The bisectors of two exterior angles and the bisector of the other interior angle are concurrent. Three intersection points, each of an external angle bisector with the opposite extended side, are collinear (fall on the same line as each other). Three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear. The angle bisector theorem
Bisection
12462183
segments). Only one of these angles contains the third side of the triangle in its interior, and this angle is called an "interior angle" of the triangle. In the picture below, the angles "∠ABC", "∠BCA" and "∠CAB" are the three interior angles of the triangle. An "exterior angle" is formed by extending one of the sides of the triangle; the angle between the extended side and the other side is the exterior angle. In the picture, angle "∠ACD" is an exterior angle. The proof of Proposition 1.16 given by Euclid is often cited as one place where Euclid gives a
Exterior angle theorem
784926
and "angulus" (angle). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a special case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons. A convex quadrilateral is a rectangle if and only if it is any one of
Rectangle
1695155
with 60° internal angles. An equilateral quadrilateral is called a rhombus, an isotoxal polygon described by an angle α. It includes the square as a special case. A convex equilateral pentagon can be described by two angles α and β, which together determine the other angles. Concave equilateral pentagons exist, as do concave equilateral polygons with any larger number of sides. An equilateral polygon which is cyclic (its vertices are on a circle) is a regular polygon (a polygon that is both equilateral and equiangular). A tangential polygon (one that has an incircle tangent to all its sides) is equilateral
Equilateral polygon
121158
equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. The sum of the angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle. The celebrated Pythagorean theorem (book I,
Euclidean geometry
12431753
the Thessalians placed their best men on the sides and the very best of these at the angles. The man at the fore angle was called an ilarches (), the one at the rear angle an ouragos () and those on the right and left angles plagiofylakes (). There are two ways to form a rhombus, one is by rank and the other by file. The first is formed with each subsequent rank containing one more trooper up to the middle rank, one less from then on. Thus, the first rank is occupied by the ilarches, the second by two
Rhombus formation
4282231
relations between the sides "s" and the angles "A" of an equilateral triangle (a triangle where all sides have the same length and all angles are equal). The relations are: Whether "C" is a right angle or not, the following relationships hold: The hyperbolic law of cosines is as follows: Its dual theorem is There is also a "law of sines": and a four-parts formula: For hyperbolic trigonometry: Hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called "sides" or "edges" and three points called "angles" or "vertices".
Hyperbolic triangle
18329946
5). The two oblique Heron triangles that share the smallest area are the acute one with sides (6, 5, 5) and the obtuse one with sides (8, 5, 5), the area of each being 12. Acute and obtuse triangles An acute triangle is a triangle with all three angles acute (less than 90°). An obtuse triangle is one with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180°, no triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles—triangles that are
Acute and obtuse triangles
1063910
the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the "ambiguous case") and the technique gives two possible values for the enclosed angle. The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions
Law of sines
825571
of what a right angle is, namely two straight lines intersecting to form two equal and adjacent angles. The straight lines which form right angles are called perpendicular. Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle). Two angles are called complementary if their sum is a right angle. Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with. Euclid's commentator Proclus gave a
Right angle
13509134
used if the quadrilateral is a right kite, since the denominator is zero in that case. If "M" and "N" are the midpoints of the diagonals, and "E" and "F" are the intersection points of the extensions of opposite sides, then the area of a bicentric quadrilateral is given by where "I" is the center of the incircle. The area of a bicentric quadrilateral can be expressed in terms of two opposite sides and the angle "θ" between the diagonals according to In terms of two adjacent angles and the radius "r" of the incircle, the area is given by
Bicentric quadrilateral
340500
terms of the sides and two opposite angles: where the sides in sequence are "a", "b", "c", "d", where "s" is the semiperimeter, and "A" and "C" are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral when . Another area formula in terms of the sides and angles, with angle "C" being between sides "b" and "c", and "A" being between sides "a" and "d", is In the case of a cyclic quadrilateral, the latter formula becomes formula_7 In a parallelogram, where both pairs of opposite sides and angles
Quadrilateral
7030543
ordinary gear is the shape of a cylinder. The pitch angle of a gear is the angle between the face of the pitch surface and the axis. The most familiar kinds of bevel gears have pitch angles of less than 90 degrees and therefore are cone-shaped. This type of bevel gear is called external because the gear teeth point outward. The pitch surfaces of meshed external bevel gears are coaxial with the gear shafts; the apexes of the two surfaces are at the point of intersection of the shaft axes. Bevel gears that have pitch angles of greater than ninety
Bevel gear
417386
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: "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. Calculating the area "T" of a triangle is an elementary problem encountered often in
Triangle
5457880
and OC are congruent, side OB is congruent to itself, and sides BA and BC are congruent. Therefore, angles OBA and OBC are equal. Finally, because they form a complete circle, we have but, due to the congruences, angle OBA = angle OBC and angle DBA = angle DBC, thus therefore points O, B, and D are collinear. Let point P be the intersection of lines AC and BD. Then, since ABCD is a rhombus, P is the midpoint of both line segments BD and AC. Therefore, length BP = length PD. Triangle BPA is congruent to triangle DPA, because
Peaucellier–Lipkin linkage
516982
Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the "hypotenuse" (side "c" in the figure). The sides adjacent to the right angle are called "legs" (or "catheti", singular: "cathetus"). Side "a" may be identified as the side "adjacent to angle B" and "opposed to" (or "opposite") "angle A", while side "b" is the side "adjacent to
Right triangle
516970
Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the "hypotenuse" (side "c" in the figure). The sides adjacent to the right angle are called "legs" (or "catheti", singular: "cathetus"). Side "a" may be identified as the side "adjacent to angle B" and "opposed to" (or "opposite") "angle A", while side "b" is the side "adjacent to
Right triangle
7789399
point is the intersection of three circles that intersect the circumcircle to form lenses with apex angle π/3. The angles formed by the first isodynamic point with the triangle vertices satisfy the equations formula_29, formula_30, and formula_31. Analogously, the angles formed by the second isodynamic point satisfy the equationsformula_32, formula_33, and formula_34. The pedal triangle of an isodynamic point, the triangle formed by dropping perpendiculars from formula_1 to each of the three sides of triangle formula_3, is equilateral, as is the triangle formed by reflecting formula_1 across each side of the triangle. Among all the equilateral triangles inscribed in triangle
Isodynamic point
825568
Right angle In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin "angulus rectus"; here "rectus" means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied
Right angle
411577
example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express. To define the trigonometric functions for the angle "A", start with any right triangle that contains the angle "A". The three sides of the triangle are named as follows: In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180° ( radians). Therefore, in a right-angled triangle, the two non-right angles total 90° ( radians), so each of these angles must be in the range
Trigonometric functions
417385
length of the other sides and the angle opposite to the unknown side. 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: If the lengths of all three sides of any triangle are known the three angles can be calculated: The law of tangents, or tangent rule, can
Triangle
5888397
with a horizontal plane. For a plane, the two angles are called its "strike (angle)" and its "dip (angle)". A "strike line" is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike angle is the "bearing" of this line (that is, relative to geographic north or from magnetic north). The dip is the angle between a horizontal plane and the observed planar feature as observed in a third vertical plane perpendicular to the strike line. The attitude of a rigid body is its orientation as described, for example, by the
Orientation (geometry)
3118031
there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases. The figure below shows the seven non-trivial cases: in each case the given sides are marked with a cross-bar and the given angles with an arc. (The given elements are also listed below the triangle). There is a full discussion of the solution of oblique triangles in Todhunter (ChapterVI). The solution methods listed here are not the only possible choices:
Spherical trigonometry
12287485
angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle "A", where "a", " b" and "c" refer to the lengths of the sides in
Trigonometry
13206616
third angle by using the angle sum property of a triangle, then find the other two sides using the law of sines. The general spherical triangle is fully determined by three of its six characteristics (3 sides and 3 angles). Note that the sides of a spherical triangle are measured by angular rather than linear units, based on the corresponding central angles. The solution of triangles for non-Euclidean spherical geometry has some differences from the planar case. For example, the sum of the three angles depends on the triangle. In addition, there are no unequal similar triangles, and so the
Solution of triangles
13206615
side length. Assume that two sides and the angle are known. The equation for the angle can be implied from the law of sines: We denote further (the equation's right side). There are four possible cases: Once is obtained, the third angle . The third side can then be found from the law of sines: or The known characteristics are the side and the angles . The third angle . Two unknown sides can be calculated from the law of sines: or The procedure for solving an AAS triangle is same as that for an ASA triangle: First, find the
Solution of triangles
8688294
of History of Science. "A remarkable synthesis of Indian spherical astronomical knowledge occurs in a passage in Tantrasamgraha." In astronomy, the spherical triangle formed by the zenith, the celestial north pole and the Sun is called the "astronomical triangle". Its sides and two of its angles are important astronomical quantities. The sides are 90° - φ where φ is the observer's terrestrial latitude, 90° - δ where δ is the Sun's declination and 90° - "a " where "a" is the Sun's altitude above the horizon. The important angles are the angle at the zenith which is the Sun's azimuth
Tantrasamgraha
5347087
plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/"l", π/"m" and π/"n" (measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those sides, 2π/"l", 2π/"m" and 2π/"n". Therefore, if the generating reflections are labeled "a", "b", "c" and the angles between them in the cyclic order are as given above, then the following relations hold: It is a theorem that all other relations between "a, b, c" are consequences
Triangle group
6044949
It is called the 'cross law' because , the square of the cosine of the angle, is called the 'cross'. For any triangle , This relation can be derived from the formula for the sine of a compound angle: in a triangle (whose three angles sum to 180°) we have, Equivalently, it describes the relationship between the spreads of three concurrent lines, as spread (like angle) is unaffected when the sides of a triangle are moved parallel to themselves to meet in a common point. Knowing two spreads allows the third to be calculated by solving the associated quadratic formula
Rational trigonometry
4282227
the third angle. The triangle where two vertices are ideal points and the remaining angle is right, one of the first hyperbolic triangles (1818) described by Ferdinand Karl Schweikart. The triangle where all vertices are ideal points, an ideal triangle is the largest possible triangle in hyperbolic geometry because of the zero sum of the angles. The relations among the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example be defined as the length of a side of an equilateral triangle with fixed angles. The length
Hyperbolic triangle
1634832
divides this segment in the ratio (when taken from the short to the long side) If the angle bisectors to angles "A" and "B" intersect at "P", and the angle bisectors to angles "C" and "D" intersect at "Q", then The term "trapezoid" was once defined as a quadrilateral without any parallel sides in Britain and elsewhere. (The Oxford English Dictionary says "Often called by English writers in the 19th century".) According to the "Oxford English Dictionary", the sense of a figure with no sides parallel is the meaning for which Proclus introduced the term "trapezoid". This is retained in
Trapezoid
14584375
inequality. An important characteristic of the automedian triangle is that the squares of its sides form an arithmetic progression. Specifically, formula_91 so formula_92. A triangle family with integer sides formula_93 and with rational bisector formula_94 of angle A is given by with integers formula_99. There exist infinitely many non-similar triangles in which the three sides and the bisectors of each of the three angles are integers. There exist infinitely many non-similar triangles in which the three sides and the two trisectors of each of the three angles are integers. However, for "n" > 3 there exist no triangles in which
Integer triangle
5731509
of the length of side "AB" to the length of side "AC": and conversely, if a point "D" on the side "BC" of triangle "ABC" divides "BC" in the same ratio as the sides "AB" and "AC", then "AD" is the angle bisector of angle "∠ A". The generalized angle bisector theorem states that if "D" lies on the line "BC", then Angles "∠ BAD" and "∠ DAC" are equal. Therefore, the right hand sides of equations () and () are equal, so their left hand sides must also be equal. which is the angle bisector theorem. If angles "∠
Angle bisector theorem
1481388
a triangle with any two sides "a" and "b", and an angle "C" between them is Each angle of an equilateral triangle is 60°, so The sine of 60° is formula_50. Thus since all sides of an equilateral triangle are equal. Equilateral triangles have frequently appeared in man made constructions: Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon,
Equilateral triangle
2125260
sides and the included angle, or two angles and a side, are known. The situation is simplified for right triangles. Consider the right triangle shown. The angle at is a right angle and the angle at is . The lengths of the sides of the triangle will be denoted as , and . can be related to the lengths of the sides as follows: The sine of , written , is defined as the ratio of the side opposite to the hypotenuse, that is, . The cosine of , written , is the side adjacent to over the hypotenuse, that
Law of tangents
5480363
axioms to Euclid's parallel postulate, has an advantage over the others: Let a line segment join the midpoint of two sides of a given triangle. That line segment will be half as long as the third side. This is equivalent to the interior angles of any triangle summing to two right angles. Given any triangle, there exists a circle that includes all of its vertices. Given any angle and any point "v" in its interior, there exists a line segment including "v", with an endpoint on each side of the angle. Begin with two triangles, "xuz" and "x'u'z'." Draw the
Tarski's axioms
2766195
equal length (properties shared with the parallelogram). The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base). Rectangles and squares are usually considered to be special cases of isosceles trapezoids though some sources would exclude them. Another special case is a "3-equal side trapezoid", sometimes known as a "trilateral trapezoid" or a "trisosceles trapezoid". They can also be seen dissected from regular polygons of 5 sides
Isosceles trapezoid
1928849
rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts: Let there be a right angle ∠ABC, r a line parallel to passing by A and s a line parallel to passing by C. Let D be the point of intersection of lines r and s (Note that it has not been proven that D lies on the circle) The quadrilateral ABCD forms a parallelogram by construction (as opposite sides are parallel). Since in a parallelogram adjacent angles are supplementary (add to 180°) and ∠ABC is a
Thales's theorem
2626218
regular polygon with sides has a circumscribed circle upon which all its vertices lie, and the center of the circle is also the center of the polygon. The central angle of the regular polygon is formed at the center by the radii to two adjacent vertices. The measure of this angle is formula_3 Central angle Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The size of a central angle
Central angle
12287487
the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are
Trigonometry
1634824
two adjacent acute angles on its longer "base" edge, while an obtuse trapezoid has one acute and one obtuse angle on each "base". An isosceles trapezoid is a trapezoid where the sides (legs) have the same length, and the base angles have the same measure. It has reflection symmetry. This is possible for acute or right trapezoids. A parallelogram is a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry). It is possible for obtuse or right trapezoids. A tangential trapezoid is a trapezoid that has an incircle. A Saccheri quadrilateral
Trapezoid
6663670
Corresponding sides and corresponding angles In geometry, the tests for congruence and similarity involve comparing corresponding sides and corresponding angles of polygons. In these tests, each side and each angle in one polygon is paired with a side or angle in the second polygon, taking care to preserve the order of adjacency. For example, if one polygon has sequential sides "a", "b", "c", "d", and "e" and the other has sequential sides "v", "w", "x", "y", and "z", and if "b" and "w" are corresponding sides, then side "a" (adjacent to "b") must correspond to either "v" or "x" (both
Corresponding sides and corresponding angles
411612
or equivalently, In this formula the angle at is opposite to the side . This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem. The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known. The following all form the law of tangents The explanation of the formulae in words would be cumbersome,
Trigonometric functions
516225
they have three identical sides and three identical angles). This can be seen as follows: One can situate one of the vertices with a given angle at the south pole and run the side with given length up the prime meridian. Knowing both angles at either end of the segment of fixed length ensures that the other two sides emanate with a uniquely determined trajectory, and thus will meet each other at a uniquely determined point; thus ASA is valid. The congruence theorems side-angle-side (SAS) and side-side-side (SSS) also hold on a sphere; in addition, if two spherical triangles have
Congruence (geometry)
516217
(also known as ASS, or Angle-Side-Side) does not by itself prove congruence. In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. There are a few possible cases: If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side (SSA, or long side-short side-angle), then the two triangles are congruent. The opposite side is sometimes longer when the corresponding angles are acute, but
Congruence (geometry)
4483610
two right angled triangles, shown here with the identical small angles in yellow. Next in sketch b we see that the two triangles have identical angles - each has a right angle, the same small angle shown in yellow, and the same larger angle shown in orange. Therefore in sketch c we see that using the principle of similar triangles, given that each triangle has identical angles, the sides will be in proportion: x the distance to the object in proportion to x', the height set on the vertical scale of the hypsometer, and h the height of the object
Hypsometer
5598274
Special right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles
Special right triangle
13206619
angle is shorter than the other side length. Known: the sides and the angle not between them. A solution exists if the following condition holds: The angle can be found from the spherical law of sines: As for the plane case, if then there are two solutions: and . We can find other characteristics by using Napier's analogies: Known: the side and the angles . First we determine the angle using the law of cosines: We can find the two unknown sides from the law of cosines (using the calculated angle ): or by using Napier's analogies: Known: the side
Solution of triangles
18329943
"x", then If two obtuse triangles have sides ("a, b, c") and ("p, q, r") with "c" and "r" being the respective longest sides, then The Calabi triangle, which is the only non-equilateral triangle for which the largest square that fits in the interior can be positioned in any of three different ways, is obtuse and isosceles with base angles 39.1320261...° and third angle 101.7359477...°. The equilateral triangle, with three 60° angles, is acute. The Morley triangle, formed from any triangle by the intersections of its adjacent angle trisectors, is equilateral and hence acute. The golden triangle is the isosceles
Acute and obtuse triangles
825569
to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry. A rectangle is a quadrilateral with four right angles. A square has four right angles, in addition to equal-length sides. The Pythagorean theorem states how to determine when a triangle is a right triangle. In Unicode, the symbol for a right angle is . It should not be confused with the similarly shaped symbol . Related symbols are , , and . In diagrams, the fact that an angle is a right angle is usually
Right angle
13206614
Note 1 above states) there is a risk of confusing an acute angle value with an obtuse one. Another method of calculating the angles from known sides is to apply the law of cotangents. Here the lengths of sides and the angle between these sides are known. The third side can be determined from the law of cosines: Now we use law of cosines to find the second angle: Finally, . This case is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other
Solution of triangles
2195500
molecule joined by a chemical bond. Every set of three not-colinear atoms of a molecule defines a plane. When two such planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify the molecular conformation. Stereochemical arrangements corresponding to angles between 0° and ±90° are called "syn" (s), those corresponding to angles between ±90° and 180° "anti" (a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called "clinal" (c) and those between 0° and ±30° or ±150° and 180° are called
Dihedral angle
411573
Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray starting at the origin and making some angle with the -axis, the sine of the angle
Trigonometric functions
12453
angles called canonical or principal angles between subspaces. In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where "U" and "V" are tangent vectors and "g" are the components of the metric tensor "G", A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the
Angle
411620
prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's "Canon triangulorum" (1620), which defines the "cosinus" as an abbreviation for the "sinus complementi" (sine of the complementary angle) and proceeds to define the "cotangens" similarly. Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In
Trigonometric functions
516215
size. (Most definitions consider congruence to be a form of similarity, although a minority require that the objects have different sizes in order to qualify as similar.) For two polygons to be congruent, they must have an equal number of sides (and hence an equal number—the same number—of vertices). Two polygons with "n" sides are congruent if and only if they each have numerically identical sequences (even if clockwise for one polygon and counterclockwise for the other) side-angle-side-angle-... for "n" sides and "n" angles. Congruence of polygons can be established graphically as follows: If at any time the step cannot
Congruence (geometry)
6069995
expression: is the angle associated with: Equation can be re-written as: Here formula_35 is an arbitrary constant that accounts for the difference in magnitude between the vectors on the two sides of the equation. The magnitudes can be ignored, only the angles are significant. Other formulas may be generated using complex numbers. For example, the angle of a complex number formula_36 is given by formula_37 and, when one multiplies complex numbers, one adds their angles. If a=b then formula_37 is 45 degrees or formula_39 radians. This means that if the real part and complex part are equal then the arctangent
Machin-like formula
417370
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 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: 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"). The triangle inequality states that the sum
Triangle
340509
"ABCD", if the angle bisectors of "A" and "C" meet on diagonal "BD", then the angle bisectors of "B" and "D" meet on diagonal "AC". The bimedians of a quadrilateral are the line segments connecting the midpoints of the opposite sides. The intersection of the bimedians is the centroid of the vertices of the quadrilateral. The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a parallelogram called the Varignon parallelogram. It has the following properties: The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that
Quadrilateral
12462185
used (see Foundations of geometry) this assertion of Euclid can be proved. The exterior angle theorem is not valid in spherical geometry nor in the related elliptical geometry. Consider a spherical triangle one of whose vertices is the North Pole and the other two lie on the equator. The sides of the triangle emanating from the North Pole (great circles of the sphere) both meet the equator at right angles, so this triangle has an exterior angle that is equal to a remote interior angle. The other interior angle (at the North Pole) can be made larger than 90°, further
Exterior angle theorem
12287476
Trigonometry Trigonometry (from Greek "trigōnon", "triangle" and "metron", "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be
Trigonometry
825473
Perpendicular In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be
Perpendicular
383912
Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles (are orthogonal) if and only if the squares of the distance between their centers is equal to the sum of the squares of their radii. If and are the equations of two distinct spheres then is also the equation of a sphere for arbitrary values of the parameters and . The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a
Sphere
3877290
Pythagorean trigonometric identity The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is given by the formula: (Note that means . This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless
Pythagorean trigonometric identity
10390133
index, the larger the deviation angle. Angle of the prism: The larger the prism angle, the larger the deviation angle. Angle of incidence: The deviation angle depends on the angle that the beam enters the object, called angle of incidence. The deviation angle first increases with increasing incidence angle, and then it decreases. There is an angle of incidence at which the sum of the two deflections is a minimum. The deviation angle at this point is called the "minimum deviation" angle, or "angle of minimum deviation". At the minimum deviation angle, the incidence and exit angles of the ray
Minimum deviation
3877300
connection with Euclid's demonstration of the Pythagorean theorem. Pythagorean trigonometric identity The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is given by the formula: (Note that means . This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same
Pythagorean trigonometric identity
8515860
Hexadecagon In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. A "regular hexadecagon" is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}. As 16 = 2 (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians. Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any
Hexadecagon
4220258
reflection and maximum gain. The internal reflections in straight-sided horns come from the two locations along the wave path where the impedance changes abruptly; the mouth or aperture of the horn, and the throat where the sides begin to flare out. The amount of reflection at these two sites varies with the "flare angle" of the horn (the angle the sides make with the axis). In narrow horns with small flare angles most of the reflection occurs at the mouth of the horn. The gain of the antenna is low because the small mouth approximates an open-ended waveguide. As the
Horn antenna
12520956
Flippin–Lodge angle The Flippin–Lodge angle is one of two angles used by organic and biological chemists studying the relationship between a molecule's chemical structure and ways that it reacts, for reactions involving "attack" of an electron-rich reacting species, the nucleophile, on an electron-poor reacting species, the electrophile. Specifically, the angles—the Bürgi–Dunitz, formula_1, and the Flippin–Lodge, formula_2—describe the "trajectory" or "angle of attack" of the nucleophile as it approaches the electrophile, in particular when the latter is planar in shape. This is called a nucleophilic addition reaction and it plays a central role in the biological chemistry taking place in many
Flippin–Lodge angle
14368449
cosines: where θ is the angle between sides "a" and "b". When θ is 90 degrees (/2 radians), then cos"θ" = 0, and the formula reduces to the usual Pythagorean theorem. At any selected angle of a general triangle of sides "a, b, c", inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled "c". Inscribing the isosceles triangle forms triangle "ABD" with angle θ opposite side "a" and with side "r" along "c". A second triangle is formed with
Pythagorean theorem
2626247
"V" and whose sides pass through points "A" and "B". Draw line "OA". Angle "BOA" is a central angle; call it "θ". Lines "OV" and "OA" are both radii of the circle, so they have equal lengths. Therefore, triangle "VOA" is isosceles, so angle "BVA" (the inscribed angle) and angle "VAO" are equal; let each of them be denoted as "ψ". Angles "BOA" and "AOV" are supplementary. They add up to 180°, since line "VB" passing through "O" is a straight line. Therefore, angle "AOV" measures 180° − "θ". It is known that the three angles of a triangle add
Inscribed angle
12506814
the triangle is isosceles with the two sides incident to the angle equal, the law of cosines simplifies significantly. Namely, because , the law of cosines becomes or An analogous statement begins by taking , , , to be the areas of the four faces of a tetrahedron. Denote the dihedral angles by formula_56 etc. Then When the angle, , is small and the adjacent sides, and , are of similar length, the right hand side of the standard form of the law of cosines can lose a lot of accuracy to numerical loss of significance. In situations where this
Law of cosines
14368443
angles, the enclosed angle and the form corresponding to Pythagoras's theorem is regained: formula_34 The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles. In a right triangle with sides "a", "b" and hypotenuse "c", trigonometry determines the sine and cosine of the angle "θ" between side "a" and the hypotenuse as: From that it follows: where the last step applies Pythagoras's theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity. In similar triangles, the ratios of the sides are the
Pythagorean theorem
9794120
holding the stone in place. Cabochons can also be set into prong settings of various kinds, but the idea is the same—it is the prongs going over the angle of the stone that creates the pressure that holds the stone in place. Just as the angle of the sides of a cabochon creates the pressure to hold the stone in place, so there is an overlying principle in setting faceted stones. If one looks at a side view of a round diamond, for example, one will see that there is an outer edge, called the girdle, and the top angles
Stonesetting
5598278
are in the ratio 1 : 2 : 3 and respectively measure 30° (), 60° (), and 90° (). The sides are in the ratio 1 : : 2. The proof of this fact is clear using trigonometry. The geometric proof is: The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if "α", , are the angles in the progression then the sum of the angles = 180°. After dividing by 3, the angle must be 60°. The right angle
Special right triangle
3226519
equal sides. Pick an arbitrary point D on side AB and construct E on AC so that AD=AE. Draw the lines BE, DC and DE. Consider the triangles BAE and CAD; BA=CA, AE=AD, and angle A is equal to itself, so by side-angle-side, the triangles are congruent and corresponding sides and angles are equal. Therefore angle ABE = angle ACD, angle ADC = angle AEB, and BE=CD. Since AB=AC and AD=AE, BD=CE by subtraction of equal parts. Now consider the triangles DBE and ECD; BD=CE, BE=CD, and angle DBE = angle ECD have just been shown, so applying side-angle-side again,
Pons asinorum
340511
the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence and Note that the two opposite sides in these formulas are not the two that the bimedian connects. In a convex quadrilateral, there is the following dual connection between the bimedians and the diagonals: The four angles of a simple quadrilateral "ABCD" satisfy the following identities: and Also, In the last two formulas, no angle is allowed to be a right angle, since tan 90° is not defined. If a convex quadrilateral has the consecutive sides "a", "b", "c", "d" and the
Quadrilateral