If the vertices of a triangle have integral coordinates
Asked by admin @ in Math viewed by 350 People
if the vertices of a triangle have integral coordinates, prove that the triangle cannot be equilateral.
Answered by admin @
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 rationalSlope 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 degreeAs we cannot find any rational point on the kline at 60 degrees so getting an equilateral triangle is not possibleHence proved
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