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 ASA_S of the polygon, the number of boundary points BB of the polygon, and the number of interior points II of the polygon is AS=I+12B1A_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. The area of the shaded triangle is 1.

If ATA_T is the area of the polygon, BB is the number of boundary points of the polygon, and II is the number of interior points of the polygon, then the relationship is AT=aI+bB+c.A_T = aI + bB + c.

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

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