Consider the triangle with vertices (0, 0), (12, 0) and (0, 9). How many points with both coordinates integers lie strictly in its interior?

Use Pick's theorem: Area = I + B/2 − 1, so I = Area − B/2 + 1.

Area of the triangle = (1/2)(12)(9) = 54.

Boundary lattice points B = gcd(12,0) + gcd(12,9) + gcd(0,9) = 12 + 3 + 9 = 24 (gcd of the coordinate differences along each side).

Interior points I = 54 − 24/2 + 1 = 54 − 12 + 1 = 43.

Tip: For a right triangle with legs on the axes of lengths p and q, boundary points = p + q + gcd(p, q).