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Prove that there cannot be a line such that it passes through all the edges of a closed and odd-sided figure made of line segments. The line cannot contain any of the vertices of the figure as one of its points.

I first set the condition that the points so chosen from the edges must be collinear and tried a bunch of cases but could not actually prove it for any specific case. Also I have no idea as to the generalized proof.

Any help would be appreciated.
Thank you.

Unaming Gaming
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    Since none of the points are on the line, if you traverse the perimeter of the polygon you will alternate between one side of the line and the other. Is that possible with an odd number of sides? –  Oct 30 '20 at 14:39
  • No it is not. I got it. Thankyou! – Unaming Gaming Oct 30 '20 at 14:51

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Every time the line you're drawing crosses an edge of the closed figure, it goes from outside to inside and vice versa. As there are an odd number of edges in the figure, the line drawn starts inside and ends outside and vice versa, which is impossible (the drawn line starts and ends outside). Thus the statement is proved.

Parcly Taxel
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