RMM 2024 Problem 5

Let $B C$ be a fixed segment in the plane, and let $A$ be a variable point in the plane not on the line $B C$. Distinct points $X$ and $Y$ are chosen on the rays $\overrightarrow{C A}$ and $\overrightarrow{B A}$, respectively, such that $\angle C B X=\angle Y C B=\angle B A C$. Assume that the tangents to the circumcircle of $A B C$ at $B$ and $C$ meet line $X Y$ at $P$ and $Q$, respectively, such that the points $X, P, Y$, and $Q$ are pairwise distinct and lie on the same side of $B C$. Let $\Omega_{1}$ be the circle through $X$ and $P$ centred on $B C$. Similarly, let $\Omega_{2}$ be the circle through $Y$ and $Q$ centred on $B C$. Prove that $\Omega_{1}$ and $\Omega_{2}$ intersect at two fixed points as $A$ varies.