Question 1
Let $\triangle ABC$ be a triangle with $AB \lt AC$, and let $D$ be a point on the side $AC$ such that $AD = \frac{AC-AB}{2}$. Points $X$ and $Y$ are chosen on the line through $A$ parallel to $BC$ such that $BX = CY$ and line $AC$ is tangent to the circumcircle of $\triangle XDY$. Prove that the tangents to the circumcircle of $\triangle XDY$ at points $X$ and $Y$ meet on line $BC$.