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

Let Ax1,y1,Bx2,y2 and  Cx3,y3  be the vertices of an equilateral triangle ABC such that x1,x2 and y1,y2 are integers. If we assume that none of the coordinates of the vertex C are irrational, then we find thatΔ= Area of ΔABC⇒ Δ=12x1    y1    1x2    y2    1x3    y3    1=A rational numberBut, Δ=34( Side )2=34×A rational number⇒ Δ= an irrational number.Thus, we arrive at a contradiction. Therefore, our supposition is wrong. Hence, at least one coordinate of C is irrational