Problem

Source: Romania TST 2022

Tags: geometry, romania, Romanian TST



Let $ABC$ be an acute triangle and let $B'$ and $C'$ be the feet of the heights $B$ and $C$ of triangle $ABC$ respectively. Let $B_A'$ and $B_C'$ be reflections of $B'$ with respect to the lines $BC$ and $AB$, respectively. The circle $BB_A'B_C'$, centered in $O_B$, intersects the line $AB$ in $X_B$ for the second time. The points $C_A', C_B', O_C, X_C$ are defined analogously, by replacing the pair $(B, B')$ with the pair $(C, C')$. Show that $O_BX_B$ and $O_CX_C$ are parallel.