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. Thanks.
Take two consecutive sides of your regular polygon, as $AB$ and $BC$ in the 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 $$ \angle OBA\cong\angle OBC\cong{1\over2}\angle ABC \cong{1\over2}\angle BCD\cong\angle OCB\cong\angle OCD. $$
Consider now side $CD$, consecutive to $BC$. By SAS triangles $BCO$ and $DCO$ are congruent, hence $DO\cong CO$ and $\angle OCD\cong\angle ODC\cong\angle ODE$. You can go on like that, to show that all vertices have the same distance from $O$.