# 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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