KZ-MO 2003 Problem 6

Let the point $B$ lie on the circle $S_1$ and let the point $A$ , other than the point $B$ , lie on the tangent to the circle $S_1$ passing through the point $B$ . Let a point $C$ be chosen outside the circle $S_1$ , so that the segment $AC$ intersects $S_1$ at two different points. Let the circle $S_2$ touch the line $AC$ at the point $C$ and the circle $S_1$ at the point $D$ , on the opposite side from the point $B$ with respect to the line $AC$ . Prove that the center of the circumcircle of triangle $BCD$ lies on the circumcircle of triangle $ABC$ .