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For Steiner $n$-chains of circles of radii $r_1,\dots,r_n$ tangent to an inner circle of radius $r_-$ and an outer circle of radius $r_+$, is there a Soddy-type relation between the $n+2$ quantities $r_1,\dots,r_n$,$r_-$, and $r_+$?


   SteinerChain
   (Image from Wikipedia added by J.O'Rourke)

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  • $\begingroup$ Presumably the answer would be of the form $P_n(r_1,\dots,r_n,r_-,r_+)=0$ where $P_n$ is a homogeneous polynomial that is cyclically symmetric (mod $n$) in $r_1,\dots,r_n$. Perhaps an algebraic geometry argument tells us ahead of time that such a polynomial must exist? To get a sense of how big the degree of $P_n$ might need to be, one could look at the situation where $r_1=r_2=\dots=r_n=1$; then $r_- = (\csc \pi/n) - 1$ and $r_+ = (\csc \pi/n) + 1$ (or something like that). Does this give information about the degree of $P_n$? $\endgroup$ – James Propp Oct 4 '13 at 3:19
  • $\begingroup$ I haven't looked at the paper, but the review of Paul Yiu, Rational Steiner porism, Forum Geom. 11 (2011) 237–249, MR2877262 suggests there might be something of interest there. $\endgroup$ – Gerry Myerson Oct 10 '13 at 22:19
  • $\begingroup$ Great question! $\endgroup$ – Gil Kalai Oct 13 '13 at 16:34
  • $\begingroup$ Maybe not generic, but $n=3$ is just given by the usual Soddy formula. Note that there are two relations - both $r_−$ and $r_+$ are determined by the others, which are free. A starting point might be to look at $n=4$, for which the middle circles are now constrained in some way (need to allow an exact incircle and circumcircle). The number of constraints appears to be $n-1$. It might be interesting to study the allowed $(r_-,r_+)$ for a given $n$. $\endgroup$ – user25199 Oct 16 '13 at 10:48
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Section 8 of this paper should answer your question.

http://cccg.ca/proceedings/2008/paper04full.pdf

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