If two vertices of an equilateral triangle have integral coordinates then the third vertex will have

Let the vertices of the equilateral triangle be (x₁, y₁), (x₂, y₂) and (x₃, y₃). 

If none of x_i and y_i (i = 1, 2, 3) are irrational, then 

area of $∆=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=retional$
But the area of an equilateral triangle
$=\frac{\sqrt{3}}{4}{\left(side\right)}^2=irrational$
Thus, the two statements are contradictory. Therefore, both the coordinates of the third vertex cannot be rational.