2014 USAMO 第 5 题

Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.

设 $ABC$ 为垂心为 $H$ 的三角形，设 $P$ 为三角形 $AHC$ 的外接圆与角 $\angle BAC$ 的内平分线的第二个交点。设$X$为三角形$APB$的外心，$Y$为三角形$APC$的垂心。证明线段$XY$的长度等于三角形$ABC$的外接圆半径。