Question 14
In triangle $ABC,$ $AB = 13,$ $BC = 15,$ and $CA = 14.$ Point $D$ is on $\overline{BC}$ with $CD = 6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE \cong \angle CAD.$ Given that $BE = \frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers, find $q.$