# Strange point in a strange polygon

Geometry Level 5

We have a regular polygon $$A$$ with vertices $$A_0, A_1, ..., A_{n-1}$$, and let $$L$$ be its side. Now, let $$P$$ be a point anywhere on the inscribed circumference of $$A$$.

If we know that $$\overline{A_0P}^2+\overline{A_1P}^2+\cdots+\overline{A_{n-1}P}^2=(45+24\sqrt{3})L^2$$, find $$n$$, i.e., the number of sides of $$A$$.

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