1998 USAMO 第 2 题

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

令${\cal C}_1$ 和${\cal C}_2$ 为同心圆，${\cal C}_2$ 位于${\cal C}_1$ 的内部。从 ${\cal C}_1$ 上的点 $A$ 绘制到 ${\cal C}_2$ 的切线 $AB$($B\in {\cal C}_2$)。令$C$ 为$AB$ 和${\cal C}_1$ 的第二个交点，并令$D$ 为$AB$ 的中点。穿过 $A$ 的直线在 $E$ 和 $F$ 处与 ${\cal C}_2$ 相交，使得 $DE$ 和 $CF$ 的垂直平分线在 $AB$ 上的点 $M$ 处相交。有证据地求出比率 $AM/MC$。