2006 APMO 第 4 题

Let $A, B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of the line segment $A B$. Let $O_{1}$ be the circle tangent to the line $A B$ at $P$ and tangent to the circle $O$. Let $\ell$ be the tangent line, different from the line $A B$, to $O_{1}$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $\ell$ and $O$. Let $Q$ be the midpoint of the line segment $B C$ and $O_{2}$ be the circle tangent to the line $B C$ at $Q$ and tangent to the line segment $A C$. Prove that the circle $O_{2}$ is tangent to the circle $O$.

令$A、B$ 为给定圆$O$ 上的两个不同点，并令$P$ 为线段$A B$ 的中点。令 $O_{1}$ 为在 $P$ 处与线 $A B$ 相切且与圆 $O$ 相切的圆。令 $\ell$ 为穿过 $A$ 到 $O_{1}$ 的切线，与线 $A B$ 不同。与$A$不同，令$C$为$\ell$和$O$的交点。设$Q$为线段$B C$的中点，$O_{2}$为在$Q$处与线$B C$相切且与线段$A C$相切的圆。证明圆 $O_{2}$ 与圆 $O$ 相切。