Is there an equilateral triangle with all its vertices in the knots of the
integral square lattice? If the answer is YES, give an example, if the answer
is NO, prove it.
You might as well assume that one vertex is at the origin, and that another one is at (p,q), for some integers p and q. The third vertex will be at a point which is the image of (p,q) under a rotation about the origin through an angle π/3 radians (or 60°). Find the coordinates of this point, and see whether they can be integers or not.