MEMO 2014 Problem 5

Let $ABC$ be a triangle with $AB < AC$ . Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$ .

Prove that $D$ is the incentre of the triangle $XYZ$ .