Question 14
Three distinct points $A$, $B$, $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose centre does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.