Number of sides of regular polygon

Question

What I need to find: Number of sides of a regular polygon

What I am given: Any 3 vertices of the polygon

What I currently know: I can find the center of the polygon. That would be the intersection of the perpendicular bisectors of any 2 lines formed by the given points.

Edit1: If there are more than one possible values of n, I am interested in finding n that would result in the minimum area polygon.

Answer 1

Let $A, B, C$ be the points on the $n$-sided polygons.

For $\Delta ABC$, denote $a=BC$, etc.

By cosine law, $$A=\cos^{-1} \left( \frac{b^{2}+c^{2}-a^{2}}{2bc} \right)$$

Angle at circumcentre $O$ correspond to arc $BC=2A$, therefore $2A,2B,2C$ should be rational multiples of $2\pi$,

Or equivalently $A=\frac{p\pi}{n},B=\frac{q\pi}{n}, C=\frac{r\pi}{n}$ where $n,p,q,r\in \mathbb{N}$ such that $p+q+r=n$

Thus $$n= \frac{1}{\gcd \left( \frac{A}{\pi},\frac{B}{\pi},\frac{C}{\pi} \right)}$$

Answer 2

Let $A$, $B$, $C$ the given points, and $O$ the center, you've constructed. Now if there is a regular polygon, that has the three points as vertices, by Euclidean algorithm you can construct the greatest common divisor of the angles $AOB\angle$, $BOC\angle$, $COA\angle$, name it $\alpha$. Still supposed, that there is a regular polygon has the vertices $A$, $B$, $C$, $\frac{2\pi}{\alpha}$ is an integer, name it $n_0$, which is the least number of the vertices of a regular polygon having the vertices $A$, $B$, $C$.

Answer 3

Assume that the three given points are vertices of a regular $n$-gon. The angle at $A$ is half the arc subtended by $BC$; therefore $\angle A={ j\over n}\pi$ for some $j\in{\mathbb N}_{\geq1}$. It follows that necessarily all three angles are rational multiples of $\pi$.

Assume, on the other hand, that $$\angle A={p\over q}\pi,\qquad \angle B={r\over s}\pi$$ in lowest terms. Then $${p\over q}={j\over n},\qquad{r\over s}={k\over n}$$ imply that $n$ has to be a multiple of $q$ as well as of $s$, hence $$n=m\ {\rm lcm}(q,s),\qquad m\in{\mathbb N}_{\geq1}\ .\tag{1}$$ On the other hand, any $n$ of the form $(1)$ gives rise to a regular $n$-gon having the three given points as vertices.