So I've noticed a couple of things about regular polygons with an even number of sides but I'm having a hard time proving them, these are all very obvious, and I think perhaps induction is the best way to prove them for any (even) n:
- The opposite sides in a regular polygon are parallel.
- Number the vertices: {1,2….2n}, if you take the side that goes from say V1 to V2, the diagonals that skip an even number of vertices, i.e., V2nV3, V2n-1V4, etc… are parallel to the given side (perhaps this can be phrased better).
- The diagonals that go from one vertex to the opposite one are concurrent.
It's quite clear in the images:
https://en.wikipedia.org/wiki/Regular_polygon#/media/File:Regular_polygon_6_annotated.svg
https://en.wikipedia.org/wiki/Regular_polygon#/media/File:Regular_polygon_12_annotated.svg
I can't seem to find this anywhere, maybe because it's too obvious to even mention it, but still thanks for any help.
Best Answer
This is evidently independent of scale and rotation. So you might as well treat the vertices as $$ v_j = (\cos (\frac{2\pi j}{2n}), \sin (\frac{2\pi j}{2n})) $$ or, in complex variables terms, $$ v_j = \exp(\frac{2\pi \mathbf i j}{2n}) $$ Once you do that, your claims should all be pretty straightforward consequences of the algebra.