Mr. Puzzle, your proof is just the analytic geometry proof. A famous theorem (called Picard theorem?) in elmentary geometry states that the area of a polygon (convex or not) equals

number of lattice points inside the polygon

+ half of number of lattice points on the boundary of the polygon

-1

For instance, the area of the triangle formed by <0, 0>, <1, 0>, and <0, 1> is 1/2.

Packman's method can be easily modified to prove this.