Middle European Mathematical Olympiad 2015 Problem T-5

Let $ABC$ be an acute triangle with $AB > AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and $$\angle AXB - \angle ACB = \angle CYA - \angle CBA$$ holds, the line $XY$ passes through $D$.