A theorem about regular polygons.

Theorem: Given a finite set of regular polygons inscribed in a circle $C$. If each pair of polygons in the set have a common vertex, then the set of vertices common to all the polygons has cardinality the greatest common divisor of the cardinalities of the vertices of the polygons in the set. The common vertices form a singleton, an antipodal pair, or are the vertices of a regular polygon inscribed in the circle.

Proof: We prove this by induction on $N$, the number of polygons in the set.
The case $N=1$ is trivially true.
Suppose the case $N=n$ is true and let $P_i,\, i=1,\cdots, n+1$ be a set of polygons satisfying the hypothesis of the theorem with each $P_i$ having $m_i$ vertices.
Let $g$ be the gcd of $m_i,\, i=1,\cdots,n$ and choose a vertex $v$ in $P_1$ which is common to $P_i,\, i=1,\cdots,n$.
Define a function $c: [0,1) \to C$ by $c(x)$ is the point $p$ on C which is $x$ of a revolution in the counterclockwise direction from $v$. So $c(j/m_i)$, $j=0,\cdots,m_i-1$ are the vertices of $P_i$ for $i=1,\cdots,n$.
And so $c(j/g=(m_i/g)\,(j/m_i))$, $j=0,\cdots,g-1$ are the common vertices of $P_i,\,i=1,\cdots,n$.
Let $g^{'}$ be the greatest common divisor of $g$ and $m_{n+1}$. So $g^{'}$ is the gcd of $m_i,\, i=1,\cdots,n+1$. Let $c(k_0/m_1)$ be a vertex of $P_{n+1}$. Then the vertices of $P_{n+1}$ are $c(k_0/m_1+i/m_{n+1}),\, i=0,\cdots,m_{n+1}-1$, where addition is addition mod 1 $a+b= f$ , with $t$ the greatest integer in $a+b$ and $f$ is the fractional remainder. So $c(k_0/g^{'}=(k_0/m_i)\,( m_i/g)\, g/g')$ for $i=1, \cdots, n+1$ is common to each polygon. Now we can write the common vertices of $P_i,\,i=1,\cdots,n+1$ as $c((k_{0}+j)/g^{'}=(k_0/m_1)(m_1/g^{'})+(j/m_{n+1})(m_{n+1}/g^{'})),\, j=0,\cdots,g^{'}-1$.$\square$

The code below allows one to choose a set of regular $m_i$ gons $i=1,\cdots,n$ to display. The first $n-1$ of them are in standard position, all sharing $(1,0)$ as a common vertex. The remaining one is rotated clockwise $rot$ of a revolution.
Press the Evaluate button and expand the cell by pressing the upper right hand corner ot the cell.

Here is an interact using the code to play with.

A question about the intersection of regular polygons.

We know that the intersection of a finite family of regular polygons insrcibed in a circle is itself a polygon. So it is the convex hull of a finite number $n$ of points in the circle. Find the minimum $m$ and maximum $M$ values for $n$.

In the interact below, you can investigate other special cases of this question. Put the polygon with the smallest number of vertices last in the list. The intersection of the polygons will be a subset of this polygon. It would interesting to know the answer even for the intersection of 2 regular polygons.