Prove that an equilateral triangle cannot have all of vertices on lattice points, i.e., points such that both are integers.
Proof. Suppose there exists such an equilateral triangle . Then,
for two disjoint, congruent right triangles . Since the vertices of are at lattice points, we know the altitude from the vertex to the base must pass through lattice points (where is the height of ). Therefore, denoting the lattice points on this altitude by , we have
But, , so,
But, , so this is a contradiction. Therefore, cannot have its vertices at lattice points and be equilateral