Question 116
Let $ABC$ be an acute triangle with circumcentre $O$. Let $\Gamma_{B}$ be the circle through $A$ and $B$ that is tangent to $AC$, and let $\Gamma_{C}$ be the circle through $A$ and $C$ that is tangent to $AB$. An arbitrary line through $A$ intersects $\Gamma_{B}$ again in $X$ and $\Gamma_{C}$ again in $Y$. Prove that $|OX| = |OY|$.