2016 IMO Shortlist G5

Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $A B C$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $A B$ and $A C$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $B C$, and let $M$ be the midpoint of $B C$. Prove that the circumcentre of triangle $X S Y$ is equidistant from $P$ and $M$.

令 $D$ 为从 $A$ 到锐角斜角三角形 $A B C$ 的欧拉线(通过外心和垂心的线)的垂直脚。以$S$为圆心的圆$\omega$穿过$A$和$D$，并分别与边$A B$和$A C$相交于$X$和$Y$。令 $P$ 为从 $A$ 到 $B C$ 的高度脚，并令 $M$ 为 $B C$ 的中点。证明三角形$X S Y$的外心与$P$和$M$等距。