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I'm doing a course in elementary euclidean geometry and I'm studying the properties of inscribed and circumscribed polygons and I've come to the following question:

For which $n$ there exist NON-regular* (1) equilateral circumscribed $n$-gons; (2) equiangular inscribed $n$-gons.

So far, not considering degenerated cases we have $n\ge 3$ and I've managed to excludes triangles and pentagons and find rhombi for (1) and rectangles for (2), however I'm not able to find a general procedure to follow.

I'm allowed to use every elementary geometrical fact (like symmetries, proportions, congruence and so on) and even a little of algebra and Cartesian geometry, however I'd prefer a coordinate-free solution.

EDIT: The definition of a regular polygon is a polygon both equilateral and equiangular, so a non-regular polygon is a polygon which is at least equilateral or equiangular and not both. A circumscribable/inscribable polygon is a polygon which can be circumscribed/inscribed by a circle.

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  • $\begingroup$ Please, recall what is a "non-regular" circumscribed polygon. $\endgroup$ – Jean Marie Jul 30 '16 at 15:41
  • $\begingroup$ @JeanMarie edited $\endgroup$ – Renato Faraone Jul 30 '16 at 15:45
  • $\begingroup$ @JeanMarie is there something wrong in my formulation? $\endgroup$ – Renato Faraone Jul 30 '16 at 15:48
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    $\begingroup$ I don't think so ! $\endgroup$ – Jean Marie Jul 30 '16 at 15:51
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It turns out that they exist if and only if $n$ is even.

Depending on your proof for the cases $n=3$ and $5$, you may have shown that the polygon must have alternating (every other) angles equal (in the equilateral case) or alternating sides (in the equiangular case) equal. If $n$ is odd, then you get every angle/side by starting with one and counting every other angle/side going around the circle, so this implies that the polygon is regular.

The converse is also true, and this gives you a construction for even $n$.

If you want to spoil the problem, take a look at this article.

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