SHARYGIN 2023 Problem 22

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$ , and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$ . The tangent to the incircle at $P$ meets $BC$ at point $X$ , and the tangent to the excircle at $Q$ meets $BC$ at $Y$ . Prove that $MX=MY$ .