Question 18
Let $ABC$ be an acute-angled triangle with $AB \ne AC$, let $H$ be its orthocenter and $M$ the midpoint of $BC$. Points $D$ on $AB$ and $E$ on $AC$ are such that $AE = AD$ and $D,H,E$ are collinear. Prove that $HM$ is orthogonal to the common chord of the circumcircles of triangles $ABC$ and $ADE$.