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Polygon X is a convex polygon which:

  • is not a triangle
  • has no pair of parallel sides
  • has all vertices with both integer coordinates
  • has sides with a length expressed by a positive integer

Can the perimeter of the polygon X be an odd number?

In Geogebra I have already drawn several dozen of such figures, but each of them had an even perimeter. I have no idea what I can do to get there.

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  • $\begingroup$ You need to use Pythagorean triplets to construct a polygon. You can start at the origin $(0, 0)$. Somehow you have to be able to close the polygon, keep it convex, and make sure no parallel sides--I don't have much advice on that. $\endgroup$
    – Jared
    Commented Dec 6, 2021 at 13:49

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I suspect you can drop "convex" and the first two constraints and it is still true that the parity of the sum is even

Consider your polygon has sides length $z_1,z_2,\ldots,z_n$ where $z_i^2=x_i^2+y_i^2$ and $\sum x_i = \sum y_i = 0$. The $x_i$ and $y_i$ are the signed differences in co-ordinates between successive vertices, corresponding to the horizontal and vertical components of each side. Everything (the lengths, the co-ordinates and the difference in co-ordinates) is an integer.

Then the parity of $z_i$ is

  • equal to the parity of $z_i^2$
  • equal to the parity of $x_i^2+y_i^2$
  • equal to the parity of $x_i+y_i$

so the parity of the circumference $\sum z_i$ is equal to the parity of $\sum x_i + \sum y_i$, which being $0$ is even.

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    $\begingroup$ [+1] Excellent. $\endgroup$
    – Jean Marie
    Commented Dec 6, 2021 at 14:07
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    $\begingroup$ @SaucyO'Path I took the $(x_i, y_i)$ to be vector components of the sides (so they add up to 0 because it's a closed polygon). $\endgroup$ Commented Dec 6, 2021 at 14:17
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    $\begingroup$ @SaucyO'Path The $x_i$ and $y_i$ are the signed differences in co-ordinates between successive vertices, corresponding to the horizontal and vertical components of each side $\endgroup$
    – Henry
    Commented Dec 6, 2021 at 14:19
  • $\begingroup$ @Henry Ah, that's right! $\endgroup$
    – user562983
    Commented Dec 6, 2021 at 14:21

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