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. 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 . 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 : 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: 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