Question 31
Let $ABC$ be an acute triangle such that $AB \lt AC$. A circle with diameter $AC$ is drawn, and on it a point $P$ such that $AP = AB$ and $P$ lies in the half-plane determined by $AC$ that does not contain $B$. $BP$ meets the circle again at $Q$, and $AQ$ meets at $R$ the line perpendicular to $BC$ through $B$. Prove that $BC$ and the bisectors of the angles $\angle BRC$ and $\angle BAC$ are concurrent.