# Generalization of some plane geometry theorems

Conjecture: Let $$A_1, A_2,\dotsc,A_n$$; $$B_1, B_2,\dotsc,B_n$$ and $$C_1, C_2,\dotsc,C_n$$ be $$3n$$ points in the plane such that $$\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2\pi}{n}$$ and $$|A_iB_i|=|A_{i+1}B_{i+1}|$$ for $$i=\overline{1,n}$$ and taking subscripts modul $$n$$. Let $$2n$$ points $$D_1, D_2,\dotsc,D_n$$; $$E_1, E_2,\dotsc,E_n$$ in the plane and $$\ell$$ is a real number 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 $$D_1D_2\dots D_n$$ is a regular $$n$$-gon $$\Leftrightarrow$$ $$E_1E_2\dots E_n$$ is a regular $$n$$-gon (in this case these two regulars $$n$$ gons have the same centroid).

This result is generalization of some results:

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

Question: Using my computer I checked the conjecture is true for $$n=3,4,5$$. Does the conjecture correct?

• mathoverflow.net/questions/366623/… Commented Jul 21, 2022 at 7:02
• What does $n+1\equiv1$ mean? What does $\ell$ stand for? Commented 2 days ago
• I am sorry! $n+1 \equiv 1$ mean: Taking subscripts modul $n$, $\ell$ is real number. I corrected. Commented 2 days ago
• In $n=3$, let $A_1, A_2, A_3$ are points $A, B, C$; $B_1, B_2, B_3$ are Fermat triangle of $ABC$. $D_1=D_2=D_3=$ the centroid of $ABC$; $C_1, C_2, C_3$ are the Midpoint of $BC, CA, AB$ $\ell=1/3$ we get Napoleon theorem Commented 2 days ago
• In $n=3$, let $A_1, A_2, A_3$ are points $A, B, C$; $B_1, B_2, B_3$ are Fermat triangle of $ABC$. $D_1=D_2=D_3=P$ is the arbitrary point in the plane of $ABC$; $C_1, C_2, C_3$ are the reflection of $A$, $B$, $C$ in $P$, $\ell=1/2$. This is one case of generalization of Dao-Nhi equilateral triangle Commented 2 days ago

Use complex coordinates (with uppercase letters replaced to corresponding lowercase) such that $$b_j-a_j=w^j$$, $$w=e^{2\pi i/n}$$. Then $$c_j-d_j=\ell(c_j-a_j)$$, so $$c_j(1-\ell)=d_j-\ell a_j$$, analogously $$c_j(1-\ell)=e_j-\ell b_j$$, thus $$d_j-\ell a_j=e_j-\ell b_j$$, $$e_j=d_j+\ell(b_j-a_j)$$, $$e_j=d_j+\ell w^j$$. $$D_1\ldots D_n$$ is regular polygon (properly directed) iff $$d_j=\alpha w^j+\beta$$ for fixed complex $$\alpha,\beta$$. Thus, we see that $$e_j$$ satisfy the same rule with $$\alpha+\ell$$ instead of $$\alpha$$.