AIME II 2026 (Problem 10)

Let

\triangle ABC

be a triangle with

D

\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}

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