I am looking for a proof (or a reference) of an inequality related to a rea and the sidelengths of a polygon as follows:

Let $A_1A_2...A_n$ be arbitrary polygon, then:

$$Area(A_1A_2....A_n) \le \frac{1}{4}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$$

• This is a generalization of Weitzenböck's inequality.

• A stronger version you can see here

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