RMO 2025 K5

Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^\circ$ and $AB < BC < AC$ . Let $M, N$ be the midpoints of $AB, AC$ respectively. Suppose $BE, CF$ are altitudes, with $E$ on $CA$ and $F$ on $AB$ . Let $X$ be the image of $M$ under reflection in the midpoint of $BF$ , and $Y$ be the image of $N$ under reflection in the midpoint of $CE$ . Prove that $XY$ bisects $BC$ .

Proposed by - Rijul Saini