CMO 1992 Problem 4

A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ . The diagonals $AC$ , $BD$ of $ABCD$ meet at $P$ . Circumcircles of $\triangle ABP$ and $\triangle CDP$ meet at $P$ and $Q$ ( $O,P,Q$ are pairwise distinct). Show that $\angle OQP=90^{\circ}$ .