CONO-SUR 2002 Problem 4

Let $ABCD$ be a convex quadrilateral such that your diagonals $AC$ and $BD$ are perpendiculars. Let $P$ be the intersection of $AC$ and $BD$ , let $M$ a midpoint of $AB$ . Prove that the quadrilateral $ABCD$ is cyclic, if and only if, the lines $PM$ and $DC$ are perpendiculars.