Question 34
Given triangle $ABC$, variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Let $Z$ be a point on the line $BC$ such that $ZX = ZY$. The circumcircle of $XYZ$ intersects the line $BC$ at $T$, for the second time. Point $P$ is chosen on line $XY$ such that $\angle PTZ = 90^\circ$. Let $Q$ be a point on the same side of line $XY$ as $A$, satisfying $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).