(This is part of an exercise from Bjarne Stroustrup's textbook - 12.12, Programming & Practice Using C++. It may have been answered somewhere, so if you know where that is, provide me the link. I'd be grateful)
I'm drawing a series of regular polygons, where each vertex of the n-sided polygon lies on a side (not sharing any vertex) of the (n+1)-sided one, n >= 3 and increased by 1 each successive polygon.
(Edit: may not be true for n >= 5. That is, "there may not exist a regular n-sided polygon where each vertex lies on a side of a (n+1)-sided polygon, for n >= 5". If so, please write a rigorous proof instead. Thank you)
Let $R_n$ be the radius of the circumcircle of a regular n-sided polygon, $(x_n, y_n)$ the co-ordinate of the circle's centre, $a_n$ the length of the polygon's side. Suppose these polygons are oriented similar to those in the illustration so that $x_n$ remains constant.
1) Does there exist a general formula (or a recursive one) to calculate the value of $R_{n+1}$ when $R_n$ is known?
2) Does there exist a formula to get $y_{n+1}$ when $y_n$ is known? Can $y_{n+1} - y_n$ be denoted only in terms of $R_n$ and/or $R_{n+1}$ and the other known constants?
Circumcircles of regular n-sided polygons, n ranges from 3 to 8
I've solved the first 3 cases where n = 3 and 4, but am stuck going further. While n = 3 is pretty straightforward, any further increase in n makes the equation become barely readable, and apparently doesn't seem to have any pattern. I would appreciate if you can provide a direction.
I notice that both R and y increase in smaller amount each successive term, which hints that they may converge to certain values. This one is just asking for curiosity. I know it's a bit too much to ask in one question, so you can just give directions, not necessarily the whole proof.
3) Do $\lim_{n\to\infty} \frac{R_n}{R_{n-1}}$ and $\lim_{n\to\infty}(y_n - y_{n-1})$ exist?