two bimedians have equal length if and only if the two diagonals are perpendicular. The two bimedians are perpendicular if and only if the two diagonals have equal length. Trigonometric identities 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. Let , , , be the sides of a convex quadrilateral, is the semiperimeter, and and are opposite angles, then and . We can use these identities to derive the Bretschneider's Formula. Inequalities Area If a convex quadrilateral has the consecutive sides a, b, c, d and the diagonals p, q, then its area K satisfies with equality only for a rectangle. with equality only for a square. with equality only if the diagonals are perpendicular and equal. with equality only for a rectangle. From Bretschneider's formula it directly follows that the area of a quadrilateral satisfies with equality if and only if the quadrilateral is cyclic or degenerate such that one side is equal to the sum of the other three (it has collapsed into a line segment, so the area is zero). The area of any quadrilateral also satisfies the inequality Denoting the perimeter as L, we have with equality only in the case of a square. The area of a convex quadrilateral also satisfies for diagonal lengths p and q, with equality if and only if the diagonals are perpendicular. Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K and diagonals AC = p, BD = q. Then with equality only for a square. Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K, then the following inequality holds: with equality only for a square. Diagonals and bimedians A corollary to Euler's quadrilateral theorem is the inequality where equality holds if and only if the quadrilateral is a parallelogram. Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that where there is equality if and only if the quadrilateral is cyclic. This is often called Ptolemy's inequality. In any convex quadrilateral the bimedians m, n and the diagonals p, q are related by the inequality with equality holding