[Pick\'s theorem] says that the area of a simple polygon P in the plane with ver

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[Pick's theorem] says that the area of a simple polygon P in the plane with vertices that are all lattice points (that is, points with integer coordinates) equals I(P)+B(P)/2-1, where I(P) and B(P) are the number of lattice points in the interior of P and on the boundary of P, respectively. Use strong induction on the number of vertices of P to prove Pick's theorem. [Hint: For the basis step, first prove the theorem for rectangles, then for rigtht triangles, and finally for all triangles by noting that the area of a triangle is the area of a larger rectangle containing it with the areas of at most three triangles subtracted. For the inductive step, take advantage of Lemma 1]

Lemma 1: Every simple polygon has an interior diagonal.

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