VMO 1995 Problem 4

ABCD is a tetrahedron. A' is the circumcenter of the face opposite A. B', C', D' are defined similarly. $p_A$ is the plane through A perpendicular to C'D', $p_B$ is the plane through B perpendicular to D'A', $p_C$ is the plane through C perpendicular to A'B', and $p_D$ is the plane through D perpendicular to B'C'. Show that the four planes have a common point. If this point is the circumcenter of ABCD, must ABCD be regular?