Tangent Spheres of Tetrahedra and a Theorem of Grace

A sphere that is tangent to all four face-planes (i.e., the affine hulls of the faces) of a tetrahedron is called a tangent sphere of the tetrahedron. Two tangent spheres are called neighboring if exactly one face-plane separates them. Grace’s theorem states that for a pair of neighboring tangent spheres S and T of a tetrahedron there is a unique sphere Θ such that (1) Θ passes through the three vertices of the tetrahedron lying on the face-plane that separates S and T, and (2) Θ is either externally tangent to both S, T or internally tangent to both S, T. It seems that this theorem is not widely known, and that no elementary proof has been given. The purpose of this article is to present an elementary and direct proof of this theorem in the case of a trirectangular tetrahedron, and to obtain several further results in this direction. Among them is also the confirmation of a slightly generalized form of Grace’s theorem.