NL-NWO 2003 Problem 4

In a circle with center $M$ , two chords $AC$ and $BD$ intersect perpendicularly. The circle of diameter $AM$ intersects the circle of diameter $BM$ besides $M$ also in point $P$ . The circle of diameter $BM$ intersects the circle with diameter $CM$ besides $M$ also in point $Q$ . The circle of diameter $CM$ intersects the circle of diameter $DM$ besides $M$ also in point $R$ . The circle of diameter $DM$ intersects the circle of diameter $AM$ besides $M$ also in point $S$ . Prove that quadrilateral $PQRS$ is a rectangle.