Equilateral Triangle Dissection

Geometry Level 5

An equilateral triangle has an interior point \(P\) from which the distances to the \(3\) vertices are distinct integers \(a\neq b\neq c\neq a\) that satisfy the condition

\(2\Delta xxx=\Delta aaa+\Delta bbb+\Delta ccc+3\Delta abc\)

where the notation \(\Delta abc\) shall denote the non-zero area of a triangle with sides \(a, b, c\), and \(x\) is the side of the equilateral triangle.

Find the minimum value \(a+b+c\) can have.


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