# Find m-n

Geometry Level 5

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 $$m-n$$.

×