1987 CMO 第 5 题

Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions:

i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$ ;

ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$ .

Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.

令 $A_1A_2A_3A_4$ 为四面体。我们构造四个相互相切的球体$S_1,S_2,S_3,S_4$，分别以$A_1,A_2,A_3,A_4$为中心。假设存在一个点$Q$，我们可以构造两个以$Q$为中心的球体，满足以下条件：

i) 一个半径为 $r$ 的球体与 $S_1,S_2,S_3,S_4$ 相切；

ii) 一个半径为 $R$ 的球体与四面体 $A_1A_2A_3A_4$ 的每条边相切。

证明 $A_1A_2A_3A_4$ 是正四面体。