A Quest for Paragon

For a want of a better term, the word paragon was used to designate a polygon whose opposite sides are parallel and equal. Following Kasner and Newman's construction of parahexagon, paragon is obtained by averaging N successive vertices of an arbitrary 2N-gon. Allowing for vertex repetition, we may as well average 3, 4, 5, etc. vertices at a time. It's almost as obvious that averaging over any number of successive vertices other than N will not in general produce a paragon. But what if we iterate the process? Choose K and form 2N averages of K successive vertices of a polygon. If need be, repeat the process. Does there exist a K, other than N, such that after a number of iterations the resulting polygon is paragon?

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