TSTST 2025 Problem 9

Let acute triangle $ABC$ have orthocenter $H$. Let $B_1$, $C_1$, $B_2$, and $C_2$ be collinear points which lie on lines $AB$, $AC$, $BH$, and $CH$, respectively. Let $\omega_B$ and $\omega_C$ be the circumcircles of triangles $BB_1B_2$ and $CC_1C_2$, respectively. Prove that the radical axis of $\omega_B$ and $\omega_C$ intersects the line through their centers on the nine-point circle of triangle $ABC$.