If vertices of triangle have integral coordinates, prove that it cannot be an equilateral triangle

Proof:

Without loss of generality we can chose the co-ordinates ( we can make this by proper shift of co-ordinates) as A= (0,0), B= (x,y), C=(a,b)

Now slope of AB = y/x is rational

Slope of AC = b/a is rational

So tan (BAC) = ((y/x) – (a/b))/ ( 1 + ay/(bx)) which is rationsl

As tan 60 = sqrt(3) there cannot be any point with rational coefficient so angle BAC cannot be 60 degree

As we cannot find any rational point on the kline at 60 degrees so getting an equilateral triangle is not possible

Hence proved