Question 14
Let the incircle of the acute triangle $ABC$ be $\omega$ and its circumcircle be $\Omega$, and let $M$ be the midpoint of side $BC$. The incircle $\omega$ touches $CA$ and $AB$ at the points $E$ and $F$, respectively, and the line $EF$ meets the circumcircle $\Omega$ at the two points $P$ and $Q$. On the circumcircle $\Gamma$ of $\triangle MPQ$ take a point $R$ such that $MR$ is perpendicular to $EF$. Prove that the line $AR$, the circle $\Gamma$ and the circle $\omega$ pass through a common point.