AIME 2012 Number 15Geometry Level 4
Triangle \(ABC\) is inscribed in circle \(\omega\) with \(AB = 5\), \(BC = 7\), and \(AC = 3\). The bisector of angle \(A\) meets side \(BC\) at \(D\) and circle \(\omega\) at a second point \(E\). Let \(\gamma\) be the circle with diameter \(DE\). Circles \(\omega\) and \( \gamma \) meet at \(E\) and a second point \(F\). Then \(AF^2 = \frac mn\), where m and n are relatively prime positive integers. Find \(m + n\).