1992 APMO 第 2 题

In a circle $C$ with centre $O$ and radius $r$, let $C_{1}, C_{2}$ be two circles with centres $O_{1}, O_{2}$ and radii $r_{1}, r_{2}$ respectively, so that each circle $C_{i}$ is internally tangent to $C$ at $A_{i}$ and so that $C_{1}, C_{2}$ are externally tangent to each other at $A$.

Prove that the three lines $O A, O_{1} A_{2}$, and $O_{2} A_{1}$ are concurrent.

在以 $O$ 为圆心、以 $r$ 为半径的圆 $C$ 中，设 $C_{1}、C_{2}$ 为两个分别以 $O_{1}、O_{2}$ 为圆心、以 $r_{1}、r_{2}$ 为半径的圆，使得每个圆 $C_{i}$ 在 $A_{i}$ 处与 $C$ 内相切，并且使 $C_{1}、C_{2}$ 与 $C_{1}、C_{2}$ 外相切彼此之间的价格为$A$。

证明 $O A、O_{1} A_{2}$ 和 $O_{2} A_{1}$ 这三行是并发的。