Question 6
Let acute triangle $ABC$ be inscribed in $(O)$ and let $D$ be on the arc $BC$ not containing $A$. A variable line $\Delta$ through the orthocenter $H$ meets the circumcircles of $ABH$ and $ACH$ again at $M\ne H$ and $N\ne H$, respectively.
(a) Determine $\Delta$ for which the area of $AMN$ is greatest.
(b) Let $d_1$ pass through $M$ perpendicular to $DB$, and $d_2$ through $N$ perpendicular to $DC$. Prove that their intersection $P$ lies on a fixed circle.
(a) Determine $\Delta$ for which the area of $AMN$ is greatest.
(b) Let $d_1$ pass through $M$ perpendicular to $DB$, and $d_2$ through $N$ perpendicular to $DC$. Prove that their intersection $P$ lies on a fixed circle.