HMMT 2025 T5

Let $\triangle ABC$ be an acute triangle with orthocenter $H$ . Points $E$ and $F$ are on segments $\overline{AC}$ and $\overline{AB}$ , respectively, such that $\angle EHF = 90^{\circ}$ . Let $X$ be the foot of the altitude from $H$ to $\overline{EF}$ . Prove that $\angle BXC = 90^{\circ}$ .