# Triangular Pick's Theorem

Geometry Level 2

Pick’s theorem states that, given a polygon with vertices on a unit square lattice, the relationship between the area $$A_S$$ of the polygon, the number of boundary points $$B$$ of the polygon, and the number of interior points $$I$$ of the polygon is $$A_S = I + \frac{1}{2}B - 1$$.

There exists a similar relationship for a polygon with vertices on an equilateral triangle lattice in which the area of the smallest possible triangle is 1.

The area of the shaded triangle is 1.

If $$A_T$$ is the area of the polygon, $$B$$ is the number of boundary points of the polygon, and $$I$$ is the number of interior points of the polygon, then the relationship is $$A_T = aI + bB + c.$$

What is $$|a|+|b|+|c|?$$

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