Question 4
Let $ABC$ be a triangle inscribed in a fixed circle $(O)$, with $BC$ fixed and $A$ varying on $(O)$. Let $H$ be the orthocenter of $ABC$. Choose $D$ on $AB$ and $E$ on $AC$ so that $H$ is the midpoint of $DE$. Prove that, as $A$ moves on $(O)$, the center of $(ADE)$ lies on a fixed circle.