If the vertices of a triangle having integral coordinates . Prove that triangle can't be equileteral .

Explanation

We have to proof that triangle can't be equilateral if the vertices of a triangle having integral coordinates.

Now let the co-ordinates as `A=(0,0),B=(x,y),C=(a,b)`

Hence, slope of `AB=y/x` is rational slope of `AC=b/a` is rational.

Therefore,

` tan(/_BAC)=(((y/x)-(a/b))/(1+(ay/bx)))` which is rational.

But at `tan(60)=sqrt(3)`, which is irrational.

implies Area of `\Delta` ABC = an irrational number.

This is a contradiction to the fact that the area is a rational number.

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