Question

prove that in an equilateral triangle, the coordinates of all the vertices cannot be rational number at the same time. ​

Answer 1

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
  
Since  is a polygon with lattice point vertices we know by the previous exercise, that . Further, by Exercise #2 of this section, we know . So,
  
But, , so,
  
But, , so this is a contradiction. Therefore,  cannot have its vertices at lattice points and be equilateral