Question 138
An equilateral triangle $ABC$ is inscribed in a circle $w$. Points $F$ and $E$ are chosen on sides $AB$ and $AC$, respectively, so that $\angle ABE + \angle ACF = 60^\circ$. The circumscribed circle of $\triangle AFE$ intersects the circle $w$ at point $D$. Rays $DE$ and $DF$ intersect line $BC$ at points $X$ and $Y$, respectively. Prove that the center of the inscribed circle of $\triangle DXY$ does not depend on the choice of points $F$ and $E$.
(Danylo Khilko)
(Danylo Khilko)