Pick's theorem says that the area of a simple polygon P in

Chapter 5, Problem 19E

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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 right 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.]