A point (x,y) in the plane is called a lattice point if both coordinates x and y are integers. Let P be a polygon whose vertices are lattice points. The area of P is I + B/2 - 1, where I denotes the number of lattice points inside the polygon and B denotes the number on the boundary.
(a) Prove that the formula is valid for rectangles with sides parallel to the coordinate axes. (b) Prove that the formula is valid for right triangles and parallelograms. (c) Use induction on the number of edges to construct a proof for general polygons.
Area of an arbitrary polygon P
P = I + B/2 - 1
where I is the number of interior points and B is the number of boundary points
The Attempt at a Solution
I have done part (a) but stuck on part (b) and (c). I have tried to represent the parallelogram as the difference between a rectangle and triangles, but no success as of now. I know I can just assume that a parallelogram has the area same as a rectangle because I have proved that it can be constructed by a rectangle and triangles, but I want to use the equation above in order to actually show that it is equal to the area of a rectangle. Any suggestions?