English: The first part of proving Pick's theorem is to show that a triangle whose vertices have integer coordinate and that contains no other integer points has area exactly 1/2. One can get a lower bound using the shoelace formula (as the coordinates are all integers, and the area is positive, we have that the determinant must be at least 1 and so the area is at least 1/2). One way to get an upper bound is to create a convex symmetric shape containing no other integer points, as then one can use Minkowski's theorem (on the interior) to show that this shape must have area at most 2^2 = 4. In this case, we can construct such a shape out of 8 triangles via translating it and a rotated copy by 180 degrees. Thus, each triangle has an area at most 4/8 = 1/2. Therefore the triangle has area 1/2.