KöMaL 2007 B4048

The vertices of a regular polygon of 2$n$ sides are $A_1, A_2, \ldots, A_{2n}$ in this order. Prove that $\frac{1}{A_1A_2^2} + \frac{1}{A_1A_n^2} = \frac{4}{A_1A_3^2}.$