2024 Mock AMC 12 - Problem 25

Triangle ABC is constructed such that \angle A = 60^\circ, AC = 8\sqrt{7}, and BC = 25. Denoting the incenter as I and orthocenter as H, the value of \sin(\angle AHI) can be expressed as \frac{m}{n} for relatively prime, positive integers m and n. Given that AB > AC, find m + n.

\textbf{(A) }7\qquad\textbf{(B) }169\qquad\textbf{(C) }196\qquad\textbf{(D) }242\qquad\textbf{(E) }243