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how do I draw a convex grid polygon with 40 sides such that it is both horizontally and vertically symmetrical and all internal angles are less than 180 and vertices lie on integer coordinates.

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One way to think about this is to have the polygon centered at $(0,0)$ and to have each quadrant contain $10$ sides. In order to have a convex polygon, the coordinates must follow some pattern where the slopes of consecutive sides either increase or decrease throughout the entire quadrant.

Below is an example of a set of coordinates that work for one quadrant: $(0,55),(10,54),(19,52),(27,49),(34,45),(40,40),(45,34),(49,27),(52,19),(54,10),(55,0)$

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  • $\begingroup$ thank you. now i get the idea that the distance of each point from(0,0) or radius will be same but how do I find out the vertices so that the coordinates are integers ? $\endgroup$ Commented Jan 13, 2021 at 7:06
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    $\begingroup$ The distances to the origin are not the same! Rather, I chose $55$ on purpose since it is the sum of the first ten positive integers. All I did was increase/decrease $x$ or $y$ by $1$ through $10$ or reversed. $\endgroup$ Commented Jan 13, 2021 at 7:35

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