2008 USAMO 第 2 题

Let $ABC$ be an acute, scalene triangle, and let $M$, $N$, and $P$ be the midpoints of $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. Let the perpendicular bisectors of $\overline{AB}$ and $\overline{AC}$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$, inside of triangle $ABC$. Prove that points $A$, $N$, $F$, and $P$ all lie on one circle.

令$ABC$为锐角不等边三角形，并令$M$、$N$和$P$分别为$\overline{BC}$、$\overline{CA}$和$\overline{AB}$的中点。设$\overline{AB}$和$\overline{AC}$的垂直平分线分别与射线$AM$相交于点$D$和$E$，并让直线$BD$和$CE$相交于三角形$ABC$内部的点$F$。证明点 $A$、$N$、$F$ 和 $P$ 都位于一个圆上。