On the existence of regular tetrahedral non-homothetic homographic solution

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).