OBM 2007 Problem 5

Let $ABCD$ be a convex quadrangle, $P$ the intersection of lines $AB$ and $CD$ , $Q$ the intersection of lines $AD$ and $BC$ and $O$ the intersection of diagonals $AC$ and $BD$ . Show that if $\angle POQ= 90^\circ$ then $PO$ is the bisector of $\angle AOD$ and $OQ$ is the bisector of $\angle AOB$ .