Question 171
Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$, and $ED$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection of the circle $(ODF)$ with $BC$.
1. Let the line through $I$ parallel to $OD$ intersect $AD$ and $ED$ at $M$ and $N$ respectively. Find the maximum value of the area of triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (with $D \neq A$).
2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$.
1. Let the line through $I$ parallel to $OD$ intersect $AD$ and $ED$ at $M$ and $N$ respectively. Find the maximum value of the area of triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (with $D \neq A$).
2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$.