1981 IMO 第 2 题

A sphere $S$ is tangent to the edges $A B, B C, C D, D A$ of a tetrahedron $A B C D$ at the points $E, F, G, H$ respectively. The points $E, F, G, H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $A C$, then it is also tangent to the edge $B D$.

球体 $S$ 与四面体 $A B C D$ 的边 $A B、B C、C D、D A$ 分别在点 $E、F、G、H$ 处相切。点 $E、F、G、H$ 是正方形的顶点。证明如果球体与边 $A C$ 相切，则它也与边 $B D$ 相切。