CWMO 2007 Problem 2

Let $C$ and $D$ be two intersection points of circle $O_1$ and circle $O_2$ . A line, passing through $D$ , intersects the circle $O_1$ and the circle $O_2$ at the points $A$ and $B$ respectively. The points $P$ and $Q$ are on circles $O_1$ and $O_2$ respectively. The lines $PD$ and $AC$ intersect at $H$ , and the lines $QD$ and $BC$ intersect at $M$ . Suppose that $O$ is the circumcenter of the triangle $ABC$ . Prove that $OD\perp MH$ if and only if $P,Q,M$ and $H$ are concyclic.