# A closed chain of $2n+1$-gon around $2n+1$-points

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 $$A_{1\;1}=O$$, $$A_{1\;2}=P_1$$, $$A_{2\;1}=A_{1\;3}$$, $$A_{2\;2}=P_{2}$$, $$A_{i+1\;1}=A_{i\;3}$$, $$A_{i+1\;2}=P_{i+1}$$ 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 $$A_{2n+1\;3}=O$$

You may easily calculate everything in complex numbers. Denote $$m=2n+1$$, $$w=e^{2\pi i/n}$$, $$Q_i=A_{i,3}=A_{i+1,2}$$ for $$i=i,2,\ldots$$. We may suppose that $$P_{m+i}=P_i$$ for $$i=1,\ldots,m$$ and we have one sequence of $$2m$$ polygons. Then we have to prove $$Q_{2m}=O$$ (let $$O=0$$ be the origin) and that $$C_k:=A_{k+m}-A_k$$ satisfy $$C_{k+1}=-w C_k$$. This follows from $$Q_k-P_k=(P_k-Q_{k-1})w$$, where $$Q_0=0$$. Dividing by $$w^k$$ this gives for the sequence $$R_k:=(-1)^kQ_k/w^k$$ the recurrence $$R_k-R_{k-1}=(-1)^kP_k(1+w)/w^k=:x_k$$. We have $$x_{m+k}+x_k=0$$, thus $$R_{2m}=x_1+\ldots+x_{2m}=0$$. Also $$A_k=(Q_k-P_kw)/(1-w)$$, thus $$C_k=A_{k+m}-A_k=\frac{1}{1-w}(Q_{k+m}-Q_k)=(-1)^{k}\frac{w^k}{1-w}(R_k+R_{m+k})= (-1)^{k}\frac{w^k}{1-w}(x_1+\ldots+x_k+x_1+\ldots+x_{m+k})= (-1)^{k}\frac{w^k}{1-w}(x_1+\ldots+x_m),$$ the result follows.
As for question 2, it asks when $$x_1+\ldots+x_m=0$$. Since $$O$$ is variable, we replace $$P_i$$ to $$P_i-O$$ and get the equation $$\sum_{k=1}^m (-1)^k (P_k-O)/w^k=0 \Leftrightarrow O=\frac{1+w}2\sum_{k=1}^m (-1)^{k-1}P_k w^{-k}.$$