Question 161
Let $k$ be a circumcircle of an acute triangle $ABC$. Consider a point $P$ on the shorter arc $BC$ of the circle $k$. Denote by $Q$ the intersection of segments $AP$ and $BC$. Let $O_1$ and $O_2$ be the circumcentres of triangles $BPQ$ and $CPQ$, respectively. Prove that if the line $O_1O_2$ passes through some vertex of the triangle $ABC$ then one of the points $O_1, O_2$ lies on the circle $k$.