Question 9
In triangle $ABC,$ $AB = 13,$ $BC = 15,$ and $CA = 17.$ Point $D$ is on $\overline{AB},$ $E$ is on $\overline{BC},$ and $F$ is on $\overline{CA}.$ Let $AD = p \cdot AB,$ $BE = q \cdot BC,$ and $CF = r \cdot CA,$ where $p,$ $q,$ and $r$ are positive and satisfy $p + q + r = \frac{2}{3}$ and $p^2 + q^2 + r^2 = \frac{2}{5}.$ The ratio of the area of triangle $DEF$ to the area of triangle $ABC$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$