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To paraphrase the title, is there such thing as an odd-sided (with at least 5 equal sides) non-regular polygon with all equal interior angles?

So for example, if I have a pentagon with all angles being $108^\circ$, then is it necessarily a regular pentagon?

I have constructed an example that I think works, but I just want to be sure.

The example I constructed is below, where you cut the red strip away from the regular pentagon and glue the halves together. The mid-line of the strip is concurrent to the line connecting the vertex to the mid-point of the opposite side.

enter image description here

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  • $\begingroup$ Sorry, I have just added conditions on $n$ to avoid that, as I forgot to do so initially. $\endgroup$
    – Trogdor
    Commented Nov 5, 2015 at 11:52
  • $\begingroup$ What example did you construct? $\endgroup$ Commented Nov 5, 2015 at 11:53
  • $\begingroup$ hexagons can have external angles all $60$ degrees but different side lengths. $\endgroup$
    – Dan Rust
    Commented Nov 5, 2015 at 11:54
  • $\begingroup$ I observed that the result was trivial for even $n$, which was why I had $n$ being odd in the title. $\endgroup$
    – Trogdor
    Commented Nov 5, 2015 at 11:57
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    $\begingroup$ Your construction is fine, but one can do more: translating any side of the $n$-gon parallel to itself does not change the angles. $\endgroup$ Commented Nov 5, 2015 at 12:00

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Here's an example of what I wrote in the above comment.

enter image description here

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  • $\begingroup$ Would never have thought of this. I think this is a much better counter example than my one, which I felt was 'cheat'-ish. $\endgroup$
    – Trogdor
    Commented Nov 5, 2015 at 12:16

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