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The point is to find some polygon for every number of sides so that a line can be drawn intersecting all of its sides and not going through its vertices.

For an even number of sides a concave kite or its extended versions prove it is possible.

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But I am not sure if it's possible for an odd number of sides, or if there is some condition by which I can say if a number of sides has such a polygon.

MyMolecules
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David K
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    Start from a point on the line far from the polygon and move towards it. Each time you cross a boundary line you pass from Exterior to Interior or conversely. If you passed through an odd number of lines, you'd be trapped in the Interior. – lulu Dec 27 '21 at 11:39
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    If there were an odd number of side then tracing it out, every vertex would be on the opposite side as previous. You endup on opposite side and no way to connect to start. – coffeemath Dec 27 '21 at 11:40
  • Does intersecting a vertex count as intersecting both sides? Or does that what "not going through its points" mean? – B. Goddard Dec 27 '21 at 11:40
  • For a proof that it's impossible for a line to intersect all sides of a triangle, see this – Vasili Dec 27 '21 at 14:48

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