, 2026

Let $ABC$ be an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on the lines $AC$ and $AB$, respectively, such that the points $B$, $C$, $D$, $E$ are concyclic, and the line $DE$ bisects the side $BC$. Suppose that the lines $BD$ and $CE$ meet at $M$. Prove that the line $HM$ is perpendicular to the symmedian through $A$.