Problem

Source: 1996 Romania NMO X p2

Tags: geometry, 3D geometry, fixed, tetrahedron



Let ABCD a tetrahedron and M a variable point on the face BCD. The line perpendicular to (BCD) in M . intersects the planes(ABC), (ACD), and (ADB) in M1, M2, and M3. Show that the sum MM1+MM2+MM3 is constant if and only if the perpendicular dropped from A to (BCD) passes through the centroid of triangle BCD.