OBM 2008 Problem 4

Let $ABCD$ be a cyclic quadrilateral and $r$ and $s$ the lines obtained reflecting $AB$ with respect to the internal bisectors of $\angle CAD$ and $\angle CBD$ , respectively. If $P$ is the intersection of $r$ and $s$ and $O$ is the center of the circumscribed circle of $ABCD$ , prove that $OP$ is perpendicular to $CD$ .