Question 15
Let $ABC$ be an acute triangle with circumcircle ($O$). $D$ is a point on arc $BC$ that does not contain $A$. A moving line $l$ through orthocenter $H$ of triangle $ABC$ cuts the circumcircles of triangle $ABH$ and triangle $ACH$ again at $M, N$ respectively ($M \neq H, N \neq H$).
a) Define the position of $l$ such that the area of $AMN$ has maximal value.
b) Denote $d_1, d_2$ be the lines through $M$ and perpendicular to $DB$, through $N$ perpendicular to $DC$ respectively. Prove that the intersection $P$ of $d_1$ and $d_2$ belongs to a fixed circle.
a) Define the position of $l$ such that the area of $AMN$ has maximal value.
b) Denote $d_1, d_2$ be the lines through $M$ and perpendicular to $DB$, through $N$ perpendicular to $DC$ respectively. Prove that the intersection $P$ of $d_1$ and $d_2$ belongs to a fixed circle.