Question

if the vertices of a triangle have integral coordinates, prove that the triangle cannot be equilateral.

Answer 1

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

Answer 2

Answer:

Step-by-step explanation:

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 possible.  

Hence proved