Let $A_1A_2\cdots A_{n}$ be a $n$ regular polygon with a circumscribed circle of radius $R$, and $P$ be the midpoint of the inferior arc ${A_1A_n}$. Prove that $$\frac{1}{PA_1^2}+\frac{1}{PA_2^2}+\cdots+\frac{1}{PA_n^2}=\frac{n^2}{4R^2}.$$
It's well known that $$PA_1^2+PA_2^2+\cdots+PA_n^2=2nR^2,$$ but this is useful for the present problem? Please give a pure geometric proof without complex numbers or other algebraic methods.