ILL 1984 Problem 62

From a point $P$ exterior to a circle $K$ , two rays are drawn intersecting $K$ in the respective pairs of points $A, A'$ and $B,B'$ . For any other pair of points $C, C'$ on $K$ , let $D$ be the point of intersection of the circumcircles of triangles $PAC$ and $PB'C'$ other than point $P$ . Similarly, let $D'$ be the point of intersection of the circumcircles of triangles $PA'C'$ and $PBC$ other than point $P$ . Prove that the points $P, D$ , and $D'$ are collinear.