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

If A 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 in the form A = xI + yB + z. What are x, y, and z?

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This is pretty much the same as the standard square lattice version. There is an affine map taking the square lattice to the triangular lattice. The only difference is that the square gets mapped to a rhombus composed of two equilateral triangles, so with area $2$ according to your convention. So Pick's theorem holds, save that the area formula is doubled.

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