Equilateral Triangle with a Point

Geometry Level 5

Given an equilateral triangle \(ABC\), and a point inside \(ABC\) as \(P\). Given, \(PA=a, PB=b, PC=c\) and \(AB=l\). We have \[l=\sqrt{\dfrac{a^2+b^2+c^2}p+q\sqrt{r}D} , \] where \(D\) is the area of the triangle formed by \(PA,PB\) and \(PC\).

It is given that \(p,q\) and \(r\) are positive integers with \(r\) square-free. Find \(pqr \).

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