Non-convex even-sided polygons whose vertices lie on a circleGiven 2 n 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?

Question

Non-convex even-sided polygons whose vertices lie on a circleGiven   2  n 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?

Eynardfb0 · Accepted Answer

Step 1Label the points   1  ,  2  ,  …  2  n in clockwise order. 2 edges       v    i        v          i      +      1       and       v    j        v          j      +      1       are parallel if and only if       v    i    +      v          i      +      1        ≡      v    j    +      v          j      +      1          (  mod    2  n  ).Step 2Suppose there is a way to connect the points so that no two edges are parallel.Then, all of these       v    i    +      v          i      +      1          (  mod    2  n  ) residues are distinct.Hence   0  ≡  (  2  n  )  (  2  n  +  1  )  ≡  2  ∑      v    i    ≡  ∑  (      v    i    +      v          i      +      1        )  ≡  n  (  2  n  +  1  )  ≡  n    (  mod    2  n  ).This is a contradiction.

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