Lattice triangles on 2d

Question

I'm trying to prove that there exist at least one almost equilateral triangle of for every error $\epsilon$>0 that has integer coordinates in the Cartesian Plane.

Answer 1

For the first part: Take a rational approximation $\frac{p}{q}$ to $\frac{\sqrt{3}}{2}=\sin(60°)$, and consider the triangle given by the coordinates $(-q,0),(0,p),(q,0)$. As $\sin$ is continuous,we can get arbritrarily close by increasing the precision of the rational approximation.