Prove that the triangle cannot be equilateral if the vertic…

Maths Rectangular Cartesian Co-ordinates Mix Subjective Type

Published on: August 11, 2026

Prove that the triangle cannot be equilateral if the vertices have integral co-ordinates

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Sol. Let A (x 1 , y 1 ), B (x 2 , y 2 ) and C (x 3 , y 3 ) be the vertices of Δ ABC, Area of triangle

Δ = 1/2 [x 1 (y 2 – y 3 ) + x 2 (y 3 – y 1 ) + x 3 (y 1 – y 2 )]

= A rational number. If possible, let the triangle ABC

be an equilateral triangle, then its area is given by

Δ = /4 (side) 2 = /4 (a positive integer)

= an irrational number

This is a contradiction to the fact that the area is a rational number. Hence the triangle can not be equilateral.

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