Editing Pick's theorem (section)

===Other proofs===
Alternative proofs of Pick's theorem that do not use Euler's formula include the following.
*One can recursively decompose the given polygon into triangles, allowing some triangles of the subdivision to have area larger than 1/2. Both the area and the counts of points used in Pick's formula add together in the same way as each other, so the truth of Pick's formula for general polygons follows from its truth for triangles. Any triangle subdivides its [[bounding box]] into the triangle itself and additional [[right triangle]]s, and the areas of both the bounding box and the right triangles are easy to compute. Combining these area computations gives Pick's formula for triangles, and combining triangles gives Pick's formula for arbitrary polygons.{{r|discretely|ball|varberg}}
*Alternatively, instead of using grid squares centered on the grid points, it is possible to use grid squares having their vertices at the grid points. These grid squares cut the given polygon into pieces, which can be rearranged (by matching up pairs of squares along each edge of the polygon) into a [[polyomino]] with the same area.{{r|trainin}}
*Pick's theorem may also be proved based on [[complex integration]] of a [[doubly periodic function]] related to [[Weierstrass's elliptic functions|Weierstrass elliptic functions]].{{r|diaz-robins}}
*Applying the [[Poisson summation formula]] to the [[characteristic function]] of the polygon leads to another proof.{{r|bcrt}}

Pick's theorem was included in a 1999 web listing of the "top 100 mathematical theorems", which later became used by Freek Wiedijk as a [[Benchmark (computing)|benchmark]] set to test the power of different [[proof assistant]]s. {{as of|2024}}, Pick's theorem had been formalized and proven in only two of the ten proof assistants recorded by Wiedijk.{{r|wiedijk}}