Disphenoid Viviani Converse Proof

Theorem: If the sum of the distances from a point P to the faces of a tetrahedron is constant, then the tetrahedron has faces of equal area (i.e. is a disphenoid).

Proof: Since the sum of distances to the faces is constant, moving point P to each of the vertices shows that the tetrahedon has four equal heights (altitudes) from each of the vertices. But since the volume is constant, irrespective from whichever face it is determined from, this implies that all the faces have the same area.