Conceptual 4

Suppose a triangle has vertices

(x,y), (a,b) and (p,q), where

x,y,a,b,p,q are all integers.

Then its area

| 1 x y |

| 1 a b | • (1/2) .................................. (1)

| 1 p q |

is a Rational Number.

But if this is an equilateral triangle,

then its area will also be = (√3 / 4 )·s²

where s is its side. Since s² is rational,

then this area, due to the presence of √3,

is irrational.

Thus, there is a contradiction as a Rational

Number equals an Irrational Number.

Hence, co-ordinates of vertices of an equilateral

triangle cannot be all integers. ..................................... Q.E.D.

Let $\omega=\dfrac 12 + \dfrac {\sqrt 3}2 i$ represent the rotation of 60 degrees.
The question is equivalent to 'is there any complex number $z$ such that both $z$ and $\omega z$ are Gaussian Integer?' which the answer is obviously no due to the $\sqrt 3$ in $\omega$ .
In fact, the answer is still no even if we are asking about rational coordinates.

Using contradiction the result can be easily proved not only for integral vertices but also for rational vertices..M-1: Use the determinant form of area of triangle using coordinates of vertices and equate it with popular formula for area of equilateral triangle.Without evaluating anything further,it can be observed that we are trying to equate a rational number with an irrational number which contradicts the supposition that the triangle is equilateral. M-2:Use the fact that slope of a line connecting two integral points must be rational and hence if we find tangent of the angle included between two such lines,it should be rational,but tan(60) is irrational.Hence proved.

If the two bottom points are on lattice points, then they are integer distance apart. For the the third point, the top, to also be on a lattice point, the triangle must have an integer height. However, if the side length of an equilateral triangle is an integer, it's height is not, since the relation of height to side length $s$ is $h= \frac{ \sqrt{3} \times s}{2}$ . Since an equilateral triangle has rotational symmetry, this thought process proves that if any two vertices of an equilateral triangle are on lattice points, the third will not be. Thus, all three vertices will never be simultaneously on lattice points.