  It is well known that the three-body problem has few analytical solutions in
certain symmetrical constraints; the Lagrangian triangular solution is one of
them. This triangular solution has been revisited by R.Broucke and H.Lass in
1971, concerning three relative position vectors pointing from one mass to
another. This paper proposes a significant advance to the method, extended to
four arbitrary masses on the vertices of a tetrahedron. The research provides a
geometrical proof that under such constraint, only homothetic solution is
possible which agrees with the conclusion brought by article 371 of Wintner
(1941).
