Školjka

%V0 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 $XYZ$.