I am looking for a proof that:
Let $A_{11}A_{12}...A_{1n}$; $A_{21}A_{22}...A_{2n}$;$\cdots$; $A_{i1}A_{i2}...A_{in}$,$\cdots$; $A_{m1}A_{m2}...A_{mn}$ are the $m$ regular polygon $n$ sides, where $n=2k$ then $Area(A_{11}A_{21}...A_{m1})+Area(A_{1\;k+1}A_{2\;k+1}...A_{m\;k+1})=Area(A_{12}A_{22}...A_{m2})+ Area(A_{1\;k+2}A_{2\;k+2}...A_{m\;k+2})=\cdots=Area(A_{1i}A_{2i}...A_{mi})+Area(A_{1\;k+i+1}A_{2\;k+i+1}...A_{m\;k+i+1})=\cdots$
Reference: