Question 15
Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5,$ $BC = 7,$ and $AC = 3.$ The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E.$ Let $\gamma$ be the circle with diameter $\overline{DE}.$ Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F.$ Then $AF^2 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$