I am looking for a solution for a conjecture as follows.

In Cartesian plane, no exist an equilateral triangle such that three vertices are integer numbers.

I hope that you like the question and let me a answer.

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Let A,B,C be the vertices.Use the fact that the area is $E=\frac{1}{2}\cdot |det(\vec{AB},\vec{AC})|$

Since $det(\vec{AB},\vec{AC})$ is an integer and the area must be of the form

$AB^2\cdot \frac{\sqrt3}{4}$ which is not an integer you can see the contradiction

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