2013 AMC 12B Problem 19

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?

\textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.