# 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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