Question 48
Let $ABC$ be an acute-angled triangle with $AC \gt BC$ and circumcircle $\omega$. Suppose that $P$ is a point on $\omega$ such that $AP = AC$ and that $P$ is an interior point of the shorter arc $BC$ of $\omega$.
Let $Q$ be the point of intersection of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA = QR$ and that $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$. Prove that the points $P, Q, R$, and $S$ are concyclic.
Let $Q$ be the point of intersection of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA = QR$ and that $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$. Prove that the points $P, Q, R$, and $S$ are concyclic.