Question 173
Let $ABCD$ be a convex quadrilateral inscribed in a circle $(O, R)$. With centers the vertices of the quadrilateral and radius $R$ we draw circles $C_A(A, R)$, $C_B(B, R)$, $C_C(C, R)$, $C_D(D, R)$. The circles $C_A$ and $C_B$ meet at $K$, the circles $C_B$ and $C_C$ meet at $L$, the circles $C_C$ and $C_D$ meet at $M$ and the circles $C_D$, $C_A$ meet at $N$. (The points $K,L,M,N$ are the second common points of the corresponding circles, given that all of them pass through the point $O$.) Prove that the quadrilateral $KLMN$ is a parallelogram.