PAMO 2001 Problem 6

Let $S_1$ be a semicircle with centre $O$ and diameter $AB$ .A circle $C_1$ with centre $P$ is drawn, tangent to $S_1$ , and tangent to $AB$ at $O$ . A semicircle $S_2$ is drawn, with centre $Q$ on $AB$ , tangent to $S_1$ and to $C_1$ . A circle $C_2$ with centre $R$ is drawn, internally tangent to $S_1$ and externally tangent to $S_2$ and $C_1$ . Prove that $OPRQ$ is a rectangle.