Let $n\leq N$ be positive integers.
We consider the problem of finding an $n$-cycle with no parallel edges in a perfect $N$-gon in the Euclidian plane.
We prove that there exist no such $n$-cycle if and only if $N=n$ and even.
And we show by construction that for every other pair
$(N,n)$, $N \geq n \geq 3$, such an $n$-cycle exists.
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