Question 49
In the plane, there are a scalene triangle $ABC$ and a point $P$ outside it. On the rays $AB$, $AC$ take points $B'$ and $C'$, respectively, such that $AB' = AC$, $AC' = AB$. Let $Q$ be the reflection of $P$ in the line $BC$. Let the circumcircles of $\triangle BB'P$ and $\triangle CC'P$ meet again at the point $P'$, and let the circumcircles of $\triangle BB'Q$ and $\triangle CC'Q$ meet again at the point $Q'$. Let $O$, $O'$ be the circumcenters of $\triangle ABC$ and $\triangle AB'C'$, respectively. Prove that
(1) the three points $O'$, $P'$, $Q'$ are collinear;
(2) $O'P' \cdot O'Q' = OA^2$.
(1) the three points $O'$, $P'$, $Q'$ are collinear;
(2) $O'P' \cdot O'Q' = OA^2$.