RU-ALLRUSSIAN 1999 Problem 117

A plane tangent to the sphere circumscribed around a tetrahedron $ABCD$ is drawn through vertex $A$. Prove that the lines of intersection of this plane with the planes of the faces $ABC$, $ACD$, and $ABD$ form six equal angles if and only if $AB \cdot CD = AC \cdot BD = AD \cdot BC$.