# Why can regular polygons be inscribed into a circumference?

Why can regular polygons be inscribed into a circumference?
Why can regular polygons be inscribed into a circumference? I have asked this to myself a lot of times. I have also wondered why all triangles can be inscribed into a circumference but I think that it is not so difficult to see why.
I also have another question. Suppose that you have a circumference of radius R and you draw a chord of length L in some direction. And then draw another circumference with the same radius and draw another chord with same length L but in another direction, is it possible to rotate the second figure or apply some transformation to it, so that it looks like the first figure? It might be a silly question but I'd appreciate any answer or help.
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Brendon Bentley
Step 1
Take two consecutive sides of your regular polygon, as AB and BC in diagram below, and draw their perpendicular bisectors, intersecting at point O. The properties of the perpendicular bisector imply $AO\cong BO\cong CO$. It follows that triangles OAB and OBC are isosceles and congruent by SSS, and $\mathrm{\angle }OBA\cong \mathrm{\angle }OBC\cong \frac{1}{2}\mathrm{\angle }ABC\cong \frac{1}{2}\mathrm{\angle }BCD\cong \mathrm{\angle }OCB\cong \mathrm{\angle }OCD.$.
Step 2
Consider now side CD, consecutive to BC. By SAS triangles BCO and BDO are congruent, hence $DO\cong CO$ and $\mathrm{\angle }OCD\cong \mathrm{\angle }ODC\cong \mathrm{\angle }ODE$. You can go on like that, to show that all vertices have the same distance from O.