Question 10
Let $\triangle ABC$ be a triangle with $D$ on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Let $\omega$ be the circle that passes through $A$ and is tangent to segment $\overline{BC}$ at $D.$ Let $E \ne A$ and $F \ne A$ be the intersections of $\omega$ with segments $\overline{AB}$ and $\overline{AC},$ respectively. Suppose that $AB = 200,$ $AC = 225,$ and all of $AE,$ $AF,$ $BD,$ and $CD$ are positive integers. Find the greatest possible value of $BC.$