Abstract

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 $\Theta$ such that (1) $\Theta$ passes through the three vertices of the tetrahedron lying on the face-plane that separates S and T, and (2) $\Theta$ 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.

摘要

与四面体的所有四个面平面（即，面的仿射外壳）相切的球称为四面体的切线球。如果恰好一个面将它们隔开，则两个切线球称为相邻切球。格雷斯定理指出，对于四面体的一对相邻切线球S和T，存在一个唯一的球体$\Theta$ 这样（1） $\Theta$通过位于面平面上的四面体的三个顶点，该顶点将S和T分开，并且（2）$\Theta$或者是外切到两个小号，Ť或内切于这两个小号，Ť。似乎该定理尚未广为人知，并且没有给出任何基本证明。本文的目的是在三矩形四面体的情况下给出该定理的基本直接证明，并在该方向上获得更多结果。其中之一是对格雷斯定理的某种广义形式的确认。