SHARYGIN 2005 Problem 94

Let $P$ be the intersection point of the diagonals of the quadrangle $ABCD$ , $M$ the intersection point of the lines connecting the midpoints of its opposite sides, $O$ the intersection point of the perpendicular bisectors of the diagonals, $H$ the intersection point of the lines connecting the orthocenters of the triangles $APD$ and $BCP$ , $APB$ and $CPD$ . Prove that $M$ is the midpoint of $OH$ .