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So I'm trying to prove the Jordan curve theorem for polygons, but I'm not sure how to show that a line segment that does not intersect the boundary has all points of the same parity. I'm also not sure how to show that all the external points are connected by a polygonal path and all the internal points are connected by a polygonal path.

lovexmango
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Here is a paper which explain the Jordan curve theorem for polygons.

aGer
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    It's very handwavy, what with all the "turning off just before" and "following along". In fact, the crucial part of any proof of the Jordan curve theorem for polygons involves such kind of path constructions. – user21820 Jul 28 '16 at 12:18