2009 IMO Shortlist G2

RUS Let $A B C$ be a triangle with circumcenter $O$. The points $P$ and $Q$ are interior points of the sides $C A$ and $A B$, respectively. The circle $k$ passes through the midpoints of the segments $B P$, $C Q$, and $P Q$. Prove that if the line $P Q$ is tangent to circle $k$ then $O P=O Q$.

RUS 设$A B C$ 为外心$O$ 的三角形。点$P$和$Q$分别是边$C A$和$A B$的内点。圆$k$穿过线段$B P$、$C Q$和$P Q$的中点。证明如果线 $P Q$ 与圆 $k$ 相切，则 $O P=O Q$。