I am looking for a proof that:

if $A_{11}A_{12}...A_{1n}$; $A_{21}A_{22}...A_{2n}$; $\cdots$; $A_{i1}A_{i2}...A_{in}$; $\cdots$; $A_{m1}A_{m2}...A_{mn}$ are $m$ regular polygons with $n$ sides, where $n=2k$, then
$$
\begin{align*}
&Area(A_{11}A_{21}...A_{m1})+Area(A_{1\;k+1}A_{2\;k+1}...A_{m\;k+1})\\=&Area(A_{12}A_{22}...A_{m2})+
Area(A_{1\;k+2}A_{2\;k+2}...A_{m\;k+2})\\=&\cdots\\=&Area(A_{1i}A_{2i}...A_{mi})+Area(A_{1\;k+i+1}A_{2\;k+i+1}...A_{m\;k+i+1})\\=&\cdots
\end{align*}
$$

**Reference:**

* [Areas que suman lo mismo](http://www.xente.mundo-r.com/ilarrosa/GeoGebra/AreasIg_Npoligonos.html)