# A Lot of Methods

Geometry Level 4

In triangle $$ABC$$, $$AB=13$$, $$BC=14$$, and $$CA=15$$. Distinct points $$D$$, $$E$$, and $$F$$ lie on segments $$\overline{BC}$$, $$\overline{CA}$$, and $$\overline{DE}$$, respectively, such that $$\overline{AD}\perp\overline{BC}$$, $$\overline{DE}\perp\overline{AC}$$, and $$\overline{AF}\perp\overline{BF}$$. The length of segment $$\overline{DF}$$ can be written as $$\frac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers. What is $$m+n$$?

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