Školjka

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 $D$.