PAGMO 2021 Problem 6

Let $ABC$ be a triangle with incenter $I$ , and $A$ -excenter $\Gamma$ . Let $A_1,B_1,C_1$ be the points of tangency of $\Gamma$ with $BC,AC$ and $AB$ , respectively. Suppose $IA_1, IB_1$ and $IC_1$ intersect $\Gamma$ for the second time at points $A_2,B_2,C_2$ , respectively. $M$ is the midpoint of segment $AA_1$ . If the intersection of $A_1B_1$ and $A_2B_2$ is $X$ , and the intersection of $A_1C_1$ and $A_2C_2$ is $Y$ , prove that $MX=MY$ .