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

Now, In my computer checked, the Inequality above can be strengthened as follows:

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

Equality holds iff $A_1A_2\cdots A_n$ is a regular n-gon.

Question: May you give your proof?