Recall, a lattice point in the plane is a point with integer coordinates. Then, we look to establish a formula for the area of the polygon,
where is the number of lattice points inside of , and is the number of lattice points on the boundary.
- Prove the formula holds for rectangles whose sides are parallel to the axes.
- Extend the result to right triangles and parallelograms whose vertices lie on lattice points.
- Prove that the formula holds for general polygons.
Proof. Let be an rectangle with sides parallel to the coordinate axes. Then, is measurable (since it is a rectangle) and .
Next, since the vertices are on lattice points, and . Thus,
Proof. We know that any right triangle can be enclosed in a rectangle with edges whose lengths are equal to the lengths of the legs of the right triangle. Further, this rectangle is composed of two congruent right triangles joined along its diagonal. These right triangles each have area one half that of the rectangle and intersect along the diagonal (which has zero area (1.7, #1) since it is a line in the plane). Given a right triangle , let be such a rectangle, and let be the right triangle that makes up the other half of , so .
Since is a rectangle we know by part (a) that
Further, any interior point of will be an interior point of either or , or will lie on their shared boundary. Thus,
where denotes the points on the (shared) hypotenuse of the two right triangles. Then, we also have for the boundary points,
Finally, since and are congruent, we know and . So, putting this all together, we have,
But, we know ; thus,
This proves the result for right triangles with vertices on lattice points
Now, for a parallelogram with vertices on lattice points, we prove that the union of two simple polygons along a single edge gives a polygon for which the formula holds if it holds for the components (since then the parallelogram can be considered as a union of right triangle and rectangles, for which we have already established that the formula holds).
Proof. Let and be polygons for which the formula holds. Then,
Further, any interior point of or will be an interior point of . Any new interior point, , must have previously been a boundary point of both and . Thus, we have the following:
(Where the is to account for the end points of the joined edge.) Thus,
Hence, the formula holds for if it holds for and . Thus, it holds for all parallelograms, as the union of right triangles and rectangles
- Proof. We already have this from part (b) since we can realize any simple polygon as the union of finitely many right triangles (i.e., every simple polygon is triangularizable)