Sure you're right, I meant the vertices are in , my mistake for no paying enough attention to wording.
If a is an interior angle of the triangle, tan(a) is what I meant by the tangent of the angle, and the proposition claims that it is rational.
In the original case of equilateral triangle, it turns out no equilateral triangle
exist with vertices in , as tan(60 deg) = sqrt(3)
That is precisely why! When speaking of triangles, we must first talk about the edges of the triangle. These edges are connected and may be interpreted as being homeomorphic to some interval in $\displaystyle \mathbb{R}$ and since any interval in $\displaystyle \mathbb{R}$ is uncountable we can't speak of lines in a countable spac such as $\displaystyle \mathbb{Q}^2$
P.S. I didn't check that my language above was precise, but you get the idea.