Consider a triangle in the plane with angles $\alpha,\beta,\gamma$ and let $G$ be the group generated by the reflections across the triangle's sides. I need to find a condition on the triangle's angles so that the group is discrete (i.e. if the orbit of any point has no accumulation points).
Name the sides of a triangle by $a,b,c$ where a is the side of the triangle opposite to vertex $\alpha$, etc. I am convinced that a necessary condition is that, if we denote by $ab$ the action of reflecting the triangle against side $b$ and then side $a$ then after some finite amount of times, say $n$ we must come back to the original triangle, i.e. $(ab)^n = e$, where $e$ is the identity action.
My problem is how to deduce here that the angle formed by sides $a,b$ (say $\alpha$) must be such that $\alpha=\pi/n$ and not say $k \cdot \pi/n$ for some $k \in \mathbb{N}$. I mean why discreteness of the group and $(ab)^n=e$ implies that we must come back necessarily in one "360 cycle" and not more?