VMO 1998 Problem 2

Let O be the circumcenter of the tetrahedron ABCD. Let A', B', C', D' be points on the circumsphere such that AA', BB', CC' and DD' are diameters. Let A" be the centroid of the triangle BCD. Define B", C", D" similarly. Show that the lines A'A", B'B", C'C", D'D" are concurrent. Suppose they meet at X. Show that the line through X and the midpoint of AB is perpendicular to CD.