Characterizations of Orthodiagonal Quadrilaterals

We prove ten necessary and sufficient conditions for a convex quadrilateral to have perpendicular diagonals. One of these is a quite new eight point circle theorem and three of them are metric conditions concerning the nonoverlapping triangles formed by the diagonals. 1. A well known characterization An orthodiagonal quadrilateral is a convex quadrilateral with perpendicular diagonals. The most well known and in problem solving useful characterization of orthodiagonal quadrilaterals is the following theorem. Five other different proofs of it was given in [19, pp.158–159], [11], [15], [2, p.136] and [4, p.91], using respectively the law of cosines, vectors, an indirect proof, a geometric locus and complex numbers. We will give a sixth proof using the Pythagorean theorem. Theorem 1. A convex quadrilateral ABCD is orthodiagonal if and only if AB + CD = BC + DA.