Question 15
In triangle $ABC,$ $AC = 13,$ $BC = 14,$ and $AB = 15.$ Points $M$ and $D$ lie on $\overline{AC}$ with $AM = MC$ and $\angle ABD = \angle DBC.$ Points $N$ and $E$ lie on $\overline{AB}$ with $AN = NB$ and $\angle ACE = \angle ECB.$ Let $P$ be the other point of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE.$ Ray $AP$ meets $\overline{BC}$ 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 $m - n.$