Suppose for $n\geq 3$ we have, $z_1,\ldots ,z_n\in\mathbb{C}$ and $|z_1|=|z_2|=\cdots=|z_n|=1$. Now I need to determine a property $P$ such that the following is true :
$\sum_{i=1}^n z_i=0\Longleftrightarrow z_1,\ldots ,z_n\mbox{ are the vertices of a polygon satisfying } P$
I have solved the cases for $n=3$ and $n=4$, in the first case, $P$ is equilateral triangle and for the second, $P$ is rectangle. But my methods does not generalize for general case. Can we solve it for general $n$ ? At least I would like to know what happens for $n=5$.