Four year ago, I proposed an inequality related to area and sides of a polygon

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

But In my computer checked, the Inequality above can be strengthen as follows:

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

Equality hold iff $A_1A_2\cdots A_n$ is a regular n-gons.

Question: May you give your proof?