Question 12
In triangle $\triangle ABC,$ $AB=13,$ $BC=14,$ 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 $m+n?$