Conjecture: Let $A_1$, $A_2$,...,$A_n$; $B_1$, $B_2$,...,$B_n$ and $C_1$, $C_2$,...,$C_n$ be $3n$ points in the plane such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2\pi}{n}$ for $i=\overline{1,n}$ and $n+1\equiv 1$. Let $2n$ points $D_1$, $D_2$,...,$D_n$; $E_1$, $E_2$,...,$E_n$ in the plane such that $\overrightarrow{C_iD_i}=\ell\overrightarrow{C_iA_i}$ and $\overrightarrow{C_iE_i}=\ell\overrightarrow{C_iB_i}$ for $i=\overline{1,n}$

then $D_1D_2...D_n$ be the regular $n$-gons $\Leftrightarrow$ $E_1E_2...E_n$ be the regular $n$-gons (in this case these two regulars $n$ gons have the same centroid).

This result is generalization of some results:

• Fermat-Dao-Nhi equilateral triangle in preamble before points X(33602).

• Napoleon theorem.

• Thébault's problem I.

Example: Let $A_1$, $A_2$,...,$A_5$; $B_1$, $B_2$,...,$B_5$ be $10$ points such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=72^0$ for $i=\overline{1,5}$. Let $D_1D_2..D_3$ be a regular pentagon in the plane. Let $C_i$ be the reflection of $A_i$ in $D_i$ and $E_i$ is the midpoint of $B_iC_i$ then $E_1E_2...E_5$ is a regular pentagon (in this example we let $n=5$, $\ell=\frac{1}{2}$

Question: Is the conjecture correct? and can we generalization this result to higher dimentions?