Let and be the coordinates of the vertices of a triangle and let be all integers. Then the area of triangle

= Some rational numbers, because x ’s and y 's are integers.

Also, if the triangle is equilateral and a be the length of its side, then A positive integer. The area of the triangle

every angle is of

Which is irrationals, because is positive integer. But earlier we have calculated that the area of the triangle is rational number. Hence it is a contradiction. Therefore, if the vertices of a triangle are integers, the triangle cannot be equilateral.