# Symmetric sum of altitudes

Geometry Level 4

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 $$m+n$$?

×