KöMaL 2011 A530

There are given 2$n$+1 points $A_1,A_3,\ldots,A_{2n+1},A_{2n},A_{2n-2},\ldots, A_2$ ($n \ge$2) lying on a circle in this order such that $\angle A_1A_2A_3 = \angle A_2A_3A_4 = \ldots = \angle A_{2n-1}A_{2n}A_{2n+1} = \frac{90^\circ}{n}$ . The lines $A_2A_3,A_3A_4,,\ldots, A_{2n-1}A_{2n}$ split the line segment $A_1 A_{2n+1}$ into 2$n$-1 part; denote their lengths by $\ell_1,\ell_2,\ldots,\ell_{2n-1}$, respectively. Prove that $\ell_1^2-\ell_2^2+\ell_3^2-\ell_4^2+-\ldots-\ell_{2n-2}^2+\ell_{2n-1}^2 = 0$.