Equilateral Triangle Dissection

Geometry Level 5

An equilateral triangle

has an interior point PP from which the distances to the 33 vertices are distinct integers abcaa\neq b\neq c\neq a that satisfy the condition

2Δxxx=Δaaa+Δbbb+Δccc+3Δabc2\Delta xxx=\Delta aaa+\Delta bbb+\Delta ccc+3\Delta abc

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

Find the minimum value a+b+ca+b+c can have.


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