BALKAN 2005 Problem 1

Let $ABC$ be an acute-angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the points of intersection of the bisectors of the angles $\angle ACB$ and $\angle ABC$ with the line $DE$ and let $Z$ be the midpoint of $BC$ . Prove that the triangle $XYZ$ is equilateral if and only if $\angle A = 60^\circ$ .