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Rahul Roy

Prove that a triangle having integral coordinates cannot be an equilateral triangle.

arunesh kumar

let the coordinates of the triangle be (a,b) , (c,d) , (e,f)
let us assume the triangle is equilateral with a side length 'l' .

now the area of triangle(A) = 1/2 | a(d-f) + c(f-b) + e(b-d) |

if the coordinates area intergral then A is integral(rational).

whereas the area of an equilateral triangle is also given by(A') = sqrt(3)/4 (l)^2

now A = A'
here, LHS is rational but RHS is irrational (bcoz' of root(3) ) wich is not possible.
so, the equilateral triangle cannot have integral coordinates..

plz correct me if i'm wrong anywhere...

Bipin Dubey

From the knowledge of coordinate geometry :

Area of triangle = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

If the coordinates are integral then the area would also be an integer.

(Side of equilateral triangle)² , a² = (y₂-y₁)² + (x₂-x₁)² which is also an integer.

Area = (

3)a²/4 which is irrational.

This is a contradiction.

So, an equilateral triangle cannot have integral coordinates.

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