UA-UMO 1998 Problem 117

Two spheres are externally tangent at point $P$ . The segments $AB$ and $CD$ touch the spheres with $A$ and $C$ lying on the first sphere and $B$ and $D$ on the second. Let $M$ and $N$ be the projections of the midpoints of segments $AC$ and $BD$ on the line connecting the centers of the spheres. Prove that $PM=PN$ .