An analytic proof of a theorem of Grace

A tangent sphere of a tetrahedron is a sphere that is tangent to all four face-planes (each being the affine hull of a face of the tetrahedron). Two tangent spheres of a tetrahedron are called neighboring if exactly one face-plane separates them. J. H. Grace proved that for any pair S, T of neighboring tangent spheres of a tetrahedron, there is a sphere passing through the three vertices (of the tetrahedron) lying on the separating face-plane of S, T that is tangent to both S, T in the same fashion (i.e., either externally tangent to both S, T or internally tangent to both S, T). His proof of this result was done by applying Lie's line-sphere transformation to Schlafli's double-six theorem for lines. The purpose of this paper is to present a proof of this result by direct calculations.