2010 AIME II Problem 15

In triangle ABC, AC=13, BC=14, and AB=15. Points M and D lie on AC with AM=MC and \angle ABD = \angle DBC. Points N and E lie on AB with AN=NB and \angle ACE = \angle ECB. Let P be the point, other than A, of intersection of the circumcircles of \triangle AMN and \triangle ADE. Ray AP meets BC at Q. The ratio \frac{BQ}{CQ} can be written in the form \frac{m}{n}, 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.