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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