The proof that $\pi/3$ is not a lattice angle is due to Lucas, while the proof that an $n$-gon does not embed in $\mathbb Z^d$ is due to Schoenberg, and has a nice proof by Scherrer. There is a AMM article on the topic you ask about "Triangles with vertices on polygonslattice points", which has some history and references, too. Also, note that except for the Euclidean approach, there is another geometric approach to lattice angles related to continued fractions. Oleg Karpenkov has some nice notes on this.
The proof that $\pi/3$ is not a lattice angle is due to Lucas, while the proof that an $n$-gon does not embed in $\mathbb Z^d$ is due to Schoenberg, and has a nice proof by Scherrer. There is a AMM article on the topic you ask about "Triangles with vertices on polygons", which has some history and references, too. Also, note that except for the Euclidean approach, there is another geometric approach to lattice angles related to continued fractions. Oleg Karpenkov has some nice notes on this.