Symmetric sum of altitudes

Geometry Level 3

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 $m+n$ ?