$\DeclareMathOperator\Area{Area}\DeclareMathOperator\cotg{cotg}$I am looking for a proof (or a reference) of an inequality related to area and the sidelengths of a polygon as follows:

> Let $A_1A_2\cdots A_n$ be an arbitrary polygon, then:
>
> $$\Area(A_1A_2\cdots A_n) \le \frac{1}{4}\cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$$

* This is a generalization of [Weitzenböck's inequality](https://en.wikipedia.org/wiki/Weitzenb%C3%B6ck%27s_inequality).

* You can see a stronger version at https://mathoverflow.net/questions/299056/strengthened-version-of-isoperimetric-inequality-with-n-polygon.

**Geometric meanings:** $$\Area(A_1A_2\cdots A_n) \le \frac{\Area(1)+\Area(2)+\dotsb +\Area(n)}{n}.$$

[![Figure illustrating geometric meaning of the inequality][1]][1]

PS: I found this inequality long time ago, that time I thought this is old inequality. But today, I think this is new because I can not see any reference for the inequality. 


  [1]: https://i.sstatic.net/tNYRv.png