CN-SMO 2008 Problem 3

In $\triangle ABC$ , side $BC>AB$ . Point $D$ lies on side $AC$ such that $\angle ABD=\angle CBD$ . Points $Q,P$ lie on line $BD$ such that $AQ\bot BD$ and $CP\bot BD$ . $M,E$ are the midpoints of side $AC$ and $BC$ respectively. Circle $O$ is the circumcircle of $\triangle PQM$ intersecting side $AC$ at $H$ . Prove that $O,H,E,M$ lie on a circle.