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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:
I can't seem to find this anywhere, maybe because it's too obvious to even mention it, but still thanks for any help.
1 Answer
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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.
