# Regular polygon determined by three vertices

Given three points being vertices of some regular polygon is it possible to find minimal number of sides of such polygon? Vertices can be chosen arbitary and it is not required for them to be adjacent. The only requirement is that they are vertices of some regular polygon.

Thanks!

Yes. Assume $A,B,C$ are vertices of a regular $n$-gon. Find the circumscribed circle of the given triangle $ABC$, with cener $O$. The angles $\angle AOB$, $\angle BOC$, $\angle COA$ are integer multiples of $\frac1n\cdot 360^\circ$. Especially, each is a rational multiple of the full angle, i.e., of the form $\frac uv\cdot 360^\circ$. The least commoon multiple of the denominators $v$ is the minimal possible $n$.