No equilateral triangle

[MM]

Prove that no three lattice points in the plane form an equilateral triangle.

[MM]

One idea:

Spoiler: show

Assume there are such points and consider them in the complex plane. Then use that if [tex]a,b,c[/tex] are the numbers to these points, then [tex]a,b,c[/tex] form equilateral triangle if [tex]a+b\omega+c\omega^2=0[/tex] where [tex]1+\omega+\omega^2=0[/tex].

[Guest]

You can see a generalization of the problem here.

[MM]

Actually I didn't know how to prove the general case, so I posted it in this forum (The Guest above is me).

[Guest]

The problem is quite hard.

[leesajohnson]

The problem is quite hard and confusing.

[Guest]

Why not......I will say I think it is quite hard also.....