2015 IMO Shortlist G6

Let $A B C$ be an acute triangle with $A B>A C$, and let $\Gamma$ be its circumcircle. Let $H$, $M$, and $F$ be the orthocenter of the triangle, the midpoint of $B C$, and the foot of the altitude from $A$, respectively. Let $Q$ and $K$ be the two points on $\Gamma$ that satisfy $\angle A Q H=90^{\circ}$ and $\angle Q K H=90^{\circ}$. Prove that the circumcircles of the triangles $K Q H$ and $K F M$ are tangent to each other. (Ukraine)

设$A B C$ 为锐角三角形，且$A B>A C$，并设$\Gamma$ 为其外接圆。设 $H$、$M$ 和 $F$ 分别为三角形的重心、$B C$ 的中点以及从 $A$ 开始的高度的脚。令$Q$和$K$为$\Gamma$上满足$\angle A Q H=90^{\circ}$和$\angle Q K H=90^{\circ}$的两个点。证明三角形 $K Q H$ 和 $K F M$ 的外接圆彼此相切。 (乌克兰)