I posed a generalization of [Theorem 3.2 In my paper](http://geometry-math-journal.ro/pdf/Volume7-Issue2/5.pdf)

**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 $A_{11}A_{12}...A_{1\;2n+1}$; ....;$A_{2n+1\;1}A_{2n+1\;2}...A_{2n+1\;2n+1}$ with center $A_1, A_2...., A_{2n+1}$ such that $O=A_{11}$, $A_{12}=A_1$, $A_{2\;1}=A_{13}$, $A_{2\;2}=A_{2}$, $A_{i+1\;1}=A_{i3}$, $A_{i+1\;2}=A_{i+1}$ for $i=\overline{1\;2n}$

[![enter image description here][1]][1]

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 $B_{11}=A_{2n+1\;3}$, $B_{12}=P_1$, $B_{2\;1}=B_{13}$, $B_{2\;2}=P_{2}$, $B_{i+1\;1}=B_{i3}$, $B_{i+1\;2}=P_{i+1}$ 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 $A_{2n+1\;3}=O$*

  [1]: https://i.sstatic.net/3er3j.png

**See also:**

* [Napoleon theorem](https://en.wikipedia.org/wiki/Napoleon%27s_theorem)

* [Van Aubel theorem](https://en.wikipedia.org/wiki/Van_Aubel%27s_theorem)

* [Petr–Douglas–Neumann theorem](https://en.wikipedia.org/wiki/Petr%E2%80%93Douglas%E2%80%93Neumann_theorem)


PS: In the conjecture, all regular polygon is same direction.