Question 41
Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $\tau_A, \tau_B$ be the tangents to $\omega$ at $A$ and $B$, which meet at $X$. Now, let $Y$ be the foot of the perpendicular from $O$ onto $CX$, and let the line through $C$ parallel to $AB$ meet $\tau_A$ at $Z$. Prove that $YZ$ bisects $AC$.