PAMO 2006 Problem 1

Let $AB$ and $CD$ be two perpendicular diameters of a circle with centre $O$ . Consider a point $M$ on the diameter $AB$ , different from $A$ and $B$ . The line $CM$ cuts the circle again at $N$ . The tangent at $N$ to the circle and the perpendicular at $M$ to $AM$ intersect at $P$ . Show that $OP = CM$ .