KöMaL 2026 B5526

Let $P$ be an interior point of circle $k$ with center $O$, different from $O$. Let $O'$ be the reflection of point $O$ across point $P$. Let $X$ be one of the intersection points of the circle with center $O'$ and radius $O'P$ and circle $k$. Line $XP$ intersects circle $k$ at point $Y$ for the second time. Prove that $P$ trisects line segment $XY$.