Consider the set Ir of points (x,y) with integer coordinates in the interior of the circle with radius r, centered at the origin, i.e. x2 + y2 < r2.
For a radius of 2, I2 contains the nine points (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1) and (1,-1). There are eight triangles having all three vertices in I2 which contain the origin in the interior. Two of them are shown below, the others are obtained from these by rotation.