2012 AIME II Problem 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 mn, where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.