I posed a generalization of Theorem 3.2 In my paper
Conjecture: Let $P_1, P_2,....,P_{2n+1}$ and $O$ be $2n+2$ points in plane. Construct a chain $2n+1$ regular ${2n+1}$-gons $A_{1\;1}A_{1\;2}...A_{1\;2n+1}$; ....;$A_{2n+1\;1}A_{2n+1\;2}...A_{2n+1\;2n+1}$ with center $A_1, A_2...., A_{2n+1}$ such that $O=A_{1\;1}$$A_{1\;1}=O$, $A_{1\;2}=A_1$$A_{1\;2}=P_1$, $A_{2\;1}=A_{1\;3}$, $A_{2\;2}=A_{2}$$A_{2\;2}=P_{2}$, $A_{i+1\;1}=A_{i3}$$A_{i+1\;1}=A_{i\;3}$, $A_{i+1\;2}=A_{i+1}$$A_{i+1\;2}=P_{i+1}$ for $i=\overline{1\;2n}$
Continuing construct a chain $2n+1$ regular ${2n+1}$-gons $B_{11}B_{12}...B_{1\;2n+1}$; ....;$B_{2n+1\;1}B_{2n+1\;2}...B_{2n+1\;2n+1}$ with centers $B_1, B_2...., B_{2n+1}$, such that $B_{1\;1}=A_{2n+1\;3}$, $B_{1\;2}=P_1$, $B_{2\;1}=B_{1\;3}$, $B_{2\;2}=P_{2}$, $B_{i+1\;1}=B_{i\;3}$, $B_{i+1\;2}=P_{i+1}$ for $i=\overline{1\;2n}$.
Then $B_{2n+1\;3}=O$ and segments $A_1B_1=A_2B_2=...=A_{2n+1}B_{2n+1}$ and $\angle (A_iB_i, A_{i+1}B_{i+1})=\frac{(2n-1)\pi}{2n+1}$
Corollary: $B_{2n+1\;3}$ is fixed point when $P_1$, $P_2$, ....,$P_n$ be moved.
Question 1: Is the conjecture correct?
Question 2: Let $P_1$, $P_2$, ....,$P_n$ are fixed point in the plane, find position of $O$ such that $A_{2n+1\;3}=O$
See also:
PS: In the conjecture, all regular polygon is same direction.