A geometry problem by Pranay Kr

In a $\triangle ABC$ , $AB=13$ , $BC=15$ , and $CA=17$ . Point $D$ is on line $AB$ , $E$ is on line $BC$ , and $F$ is on line $CA$ . Let $AD=p \times BE=q \times BC$ , and $CF=r \times CA$ , where $p$ , $q$ , and $r$ are positive and satisfy $p+q+r=\dfrac 23$ and $p^2+q^2+r^2= \dfrac 25$ . The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ can be written in the form $\dfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .