ISL 1988 Problem 53

Given $n$ points $A_1, A_2, \ldots, A_n,$ no three collinear, show that the $n$ - gon $A_1 A_2 \ldots A_n,$ is inscribed in a circle if and only if

$A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots$ $+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n$ $= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n$ ,

where $XY$ denotes the length of the segment $XY.$