Let
![A_1A_2 \ldots A_n](/media/m/3/d/b/3dbfe0f5b1f84bfc9be4ee4d64979438.png)
be a convex polygon,
![n \geq 4.](/media/m/0/3/7/03742984dbd46bbce6d0380d10a7f607.png)
Prove that
![A_1A_2 \ldots A_n](/media/m/3/d/b/3dbfe0f5b1f84bfc9be4ee4d64979438.png)
is cyclic if and only if to each vertex
![A_j](/media/m/1/c/c/1ccbd0ef9163d00926a9e83c3c1af766.png)
one can assign a pair
![(b_j, c_j)](/media/m/6/d/4/6d43f7e931103687b38a3505da6d6e9e.png)
of real numbers,
![j = 1, 2, \ldots, n,](/media/m/f/1/c/f1ca0ca4cc00471bb467dca0598c2e0d.png)
so that
![A_iA_j = b_jc_i - b_ic_j](/media/m/0/9/5/095ca7b73c498da03ca2853f4564f551.png)
for all
![i, j](/media/m/5/a/a/5aac541b853183ea20751471ec3be677.png)
with
%V0
Let $A_1A_2 \ldots A_n$ be a convex polygon, $n \geq 4.$ 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, $j = 1, 2, \ldots, n,$ so that $A_iA_j = b_jc_i - b_ic_j$ for all $i, j$ with $1 \leq i < j \leq n.$