# $\sum\limits_{i=1}^n z_i=0\Longleftrightarrow z_1,\ldots ,z_n$ are the vertices of …

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 $n=5$.

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