1969 IMO 第 4 题

Let $A B$ be a diameter of a circle $\gamma$. A point $C$ different from $A$ and $B$ is on the circle $\gamma$. Let $D$ be the projection of the point $C$ onto the line $A B$. Consider three other circles $\gamma_{1}, \gamma_{2}$, and $\gamma_{3}$ with the common tangent $A B: \gamma_{1}$ inscribed in the triangle $A B C$, and $\gamma_{2}$ and $\gamma_{3}$ tangent to both (the segment) $C D$ and $\gamma$. Prove that $\gamma_{1}, \gamma_{2}$, and $\gamma_{3}$ have two common tangents.

设$A B$ 为圆$\gamma$ 的直径。与$A$和$B$不同的点$C$位于圆$\gamma$上。令 $D$ 为点 $C$ 在直线 $A B$ 上的投影。考虑另外三个圆 $\gamma_{1}、\gamma_{2}$ 和 $\gamma_{3}$，它们的公切线 $A B: \gamma_{1}$ 内切于三角形 $A B C$，并且 $\gamma_{2}$ 和 $\gamma_{3}$ 与(线段)$C D$ 和 $\gamma$ 相切。证明 $\gamma_{1}、\gamma_{2}$ 和 $\gamma_{3}$ 有两条公切线。