Question 10
Let $\triangle ABC$ be a triangle with $\angle BAC = 2\angle CBA$ and $AB \lt AC$. A point $P$ lies on segment $AC$ such that $PC = AB + AP$. Let $O$ be the circumcentre of triangle $\triangle ABP$, and let the line through $O$ parallel to $AB$ intersect $BP$ at $Q$. Show that $AQ$ passes through the midpoint of segment $BC$.