Question 3
In the plane, let $(O)$ be a circle and let $B,C$ be fixed points on it such that chord $BC$ is not a diameter. Let $A$ vary on $(O)$ with $AB\ne AC$ and $A$ distinct from $B,C$. Let $D,E$ be the intersections of line $BC$ with the internal and external bisectors, respectively, of $\angle BAC$. Let $I$ be the midpoint of $DE$. The line through the orthocenter of triangle $ABC$ perpendicular to $AI$ meets $AD,AE$ at $M,N$, respectively.
1. Prove that line $MN$ always passes through a fixed point.
2. Determine the positions of $A$ for which triangle $AMN$ has greatest area.
1. Prove that line $MN$ always passes through a fixed point.
2. Determine the positions of $A$ for which triangle $AMN$ has greatest area.