Question 157
In the plane given a circle $(O)$ and two fixed points $B$, $C$ on the circle, such that $BC$ is not a diameter. Consider a point $A$ moving on $(O)$ in such a way that $AB = AC$ and $A$ is not coincident with $B$, $C$. Denote by $D$ and $E$ the intersections of line $BC$ with the internal bisector and the external bisector of angle $BAC$. Let $I$ be the midpoint of $DE$. The line passing through the orthocenter of $ABC$ and perpendicular with $AI$ intersects lines $AD$ and $AE$ in $M$ and $N$ respectively.
a/ Show that $MN$ always passes through a fixed point.
b/ Determine the positions of $A$ such that the triangle $ABC$ has the largest area.
a/ Show that $MN$ always passes through a fixed point.
b/ Determine the positions of $A$ such that the triangle $ABC$ has the largest area.