Question 82
Let $ABC$ be a triangle with $AB \lt AC$. Let $\omega$ be a circle passing through $B$ and $C$, and assume that the point $A$ lies inside the circle $\omega$. Let $X$ and $Y$ be points of $\omega$ such that $\widehat{BX A} = \widehat{A Y C}$. Assume that $X$ and $C$ lie on opposite sides of the line $(AB)$ and that $Y$ and $B$ lie on opposite sides of the line $(AC)$. Prove that, as $X$ and $Y$ vary on the circle $\omega$, the line $(XY)$ passes through a fixed point.