## Question

###### 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

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 $1 .]$

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