Existence of regular $n$-gon whose vertices are arbitrarily close to integer coordinates

Question

I'm self-studying these days about polytopes and I came with this question. I don't know if it's true or not.

Let $\alpha_1$, $\ldots$, $\alpha_n$ angles of convex $n$-gon, $n\not=4$. Prove that for every $\varepsilon>0$, there exists a convex $n$-gon with vertices in $\mathbb{Z}^2$ such that $|\frac{\pi}{n}(n-2)-\alpha_i|<\varepsilon$

I have thought of two possible alternatives, one of them is Dirichlet Approximation and the other is showing that $SO(2,\mathbb{Q})$ is dense in $SO(2)$.

Answer 1

Hint: Let $p_1,\ldots,p_n$ be the vertices of a regular $n$-gon in $\Bbb{R}^2$. For each $p_i$ take a point $q_i\in\Bbb{Q}^2$ that is sufficiently close to $p_i$ (depending on $\varepsilon$). Now scale up all the coordinates of the $q_i$ to get vertices in $\Bbb{Z}^2$ without changing the angles $\alpha_i$. This video illustrates the idea for $n=3$.

Answer 2

For $n=6$ it's easy. Start with a triangle. Multiply its sides by $3$. Trisect the tripled sides and render the trisection points as vertices of a hexagon. The hexagon will have three pairs of parallel and congruent sides.