In triangle A B C, A B=13, B C=14, and C A=15. Distinct points D, E, and F lie on segments \overline{B C}, \overline{C A}, and \overline{D E}, respectively, such that \overline{A D} \perp \overline{B C}, \overline{D E} \perp \overline{A C}, and \overline{A F} \perp \overline{B F}. The length of segment \overline{D F} can be written as \dfrac{m}{n}, where m and n are relatively prime positive integers. What is m+n ?

**Answer Choices**

A. 18

B. 21

C. 24

D. 27

E. 30