ILL 1978 Problem 20

Let $O$ be the center of a circle. Let $OU,OV$ be perpendicular radii of the circle. The chord $PQ$ passes through the midpoint $M$ of $UV$ . Let $W$ be a point such that $PM = PW$ , where $U, V,M,W$ are collinear. Let $R$ be a point such that $PR = MQ$ , where $R$ lies on the line $PW$ . Prove that $MR = UV$ .

Alternative version: A circle $S$ is given with center $O$ and radius $r$ . Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$ . Let $PMQ$ be a chord of $S$ . The point $N$ is defined by $\overrightarrow{PN} =\overrightarrow{MQ}$ . Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$ . Prove that $MR =\sqrt{2}r$ .