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 $$BE=\frac pq$$ where $$p$$ and $$q$$ are relatively prime positive integers, find $$q.$$

(第二十三届AIME2 2005 第14题)