TUYMAADA 2024 S4

A triangle $ABC$ is given. The segment connecting the points where the excircles touch $AB$ and $AC$ meets the bisector of angle $C$ at $X$ . The segment connecting the points where the excircles touch $BC$ and $AC$ meets the bisector of angle $A$ at $Y$ . Prove that the midpoint of $XY$ is equidistant from $A$ and $C$ .