2000 USAMO 第 5 题

Let $A_1A_2A_3$ be a triangle and let $\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots, 7,$ $\omega_k$ is externally tangent to $\omega_{k - 1}$ and passes through $A_k$ and $A_{k + 1},$ where $A_{n + 3} = A_{n}$ for all $n \ge 1$. Prove that $\omega_7 = \omega_1.$

设 $A_1A_2A_3$ 为三角形，$\omega_1$ 为穿过 $A_1$ 和 $A_2.$ 的平面内的圆。 假设存在圆 $\omega_2, \omega_3, \dots, \omega_7$，使得对于 $k = 2, 3, \dots, 7,$ $\omega_k$ 与 $\omega_{k - 1}$ 外切并穿过 $\omega_{k - 1}$ $A_k$ 和 $A_{k + 1},$，其中 $A_{n + 3} = A_{n}$ 对于所有 $n \ge 1$。证明 $\omega_7 = \omega_1.$