Question 23
G4 (RUS). Let $A_1A_2\ldots A_n$ be a convex polygon, $n\ge4$. Prove that $A_1A_2\ldots A_n$ is cyclic if and only if to each vertex $A_j$ one can assign a pair $(b_j,c_j)$ of real numbers, $j=1,2,\ldots,n$, such that
$$A_iA_j=b_jc_i-b_ic_j.$$
for all $i,j$ with $1\le i\le j\le n$.
$$A_iA_j=b_jc_i-b_ic_j.$$
for all $i,j$ with $1\le i\le j\le n$.