Symmetric sum of altitudes

Triangle $ABC$has area 15 and perimeter 20. Furthermore, the product of the 3 side lengths is 255. If the three altitudes of the triangle have lengths $d, e$, and $f$, then the value of $de+ef+fd$can be written as $\frac{m}{n}$for relatively prime positive integers $m$and $n$. What is $m+n$?