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 \(m+n\)?

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