Problem #WSP-000306

Problem

A simple polygon is a polygon that does not intersect itself and has no holes. Suppose we have a simple polygon $S$ whose vertices consists of only integer coordinates.

The area turns out to be remarkably easy to calculate. Count up the number of points with integer coordinate inside the polygon and on the boundary; call them $i$ and $b$ respectively. The area is then $$A(S)=i+\frac{b}{2}-1.$$

In the picture above, $i=3$ and $b=11$, so $A(S)=\frac{15}{2}$. Prove that this formula for the area $A(S)$ is correct.