non-convex even-sided polygons whose vertices lie on a circle

Question

Given $2n$ evenly spaced points on a circle, the opposite sides in the convex polygon (formed by these points) are parallel. If we remove the requirement of convexity, I get degenerate polygons that always have parallel sides. How does one show that a non-convex polygon (possibly degenerate) formed by an even number of evenly spaced points on a circle always has a pair of parallel edges?

Answer 1

Hint: Label the points $1, 2, \ldots 2n$ in clockwise order.
2 edges $v_i v_{i+1}$ and $v_jv_{j+1}$ are parallel if and only if $v_i + v_{i+1} \equiv v_j + v_{j+1} \pmod{2n}$.

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Suppose there is a way to connect the points so that no two edges are parallel.

Then, all of these $v_i + v_{i+1} \pmod{2n}$ residues are distinct.

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Hence $0 \equiv (2n)(2n+1) \equiv 2\sum v_i \equiv \sum (v_i + v_{i+1} ) \equiv n(2n+1) \equiv n \pmod{2n} $.

This is a contradiction.