# Shortest closed Steiner chain of circles sandwiched between circumcircle and incircle of triangle

Consider Steiner chain of circles with the external circle of radius $$R$$, the internal circle of radius $$r$$ and $$n$$ circles in a chain with the radii $$r_1,\dots,r_n$$.

Known condition for the distance $$d$$ between the centers of the boundary circles is defined by the two radii of the boundary circles and the number $$n$$ of circles in a chain, \begin{align} d^2&=(R-r)^2-4rR\tan^2\tfrac\pi{n} \tag{1}\label{1} . \end{align}

Since every triangle has an associated pair of circles, circumscribed circle with circumradius $$R$$ and inscribed circle with inradius $$r$$ (one is always inside the other), this pair of circles looks like a natural candidate to use in construction of the Steiner chain of circles.

The distance between the centers of the circumcircle and the incircle is known to obey \begin{align} d^2&=R(R-2r) \tag{2}\label{2} . \end{align}

Let for simplicity set $$R=1$$. Then relations \eqref{1} and \eqref{2} suggest that \begin{align} r&=4\tan^2\tfrac\pi{n} \tag{3}\label{3} . \end{align}

As we know that for triangles $$r$$ must be less than of equal $$\tfrac12$$, it follows that:

1) The Steiner chain of circles can not be constructed using circumcircle+incircle of the equilateral triangle (trivial to check).

2) The shortest Steiner chain of circles sandwiched between the circumcircle and the incircle of triangle consists of $$10$$ (ten) circles, all such triangles are strictly acute (for example, $$\triangle UVW$$), and there are two isosceles ($$\triangle A_1B_1C_1$$ and $$\triangle A_2B_2C_2$$ ) among them: 3) The shortest Steiner chain of circles associated with the right-angled triangle has length $$n=11$$: Question.

Are there any known references where this kind of relation between triangles and Steiner chains of circles was examined in details?

Any suggestions?

• Very interesting questions and results. I have never seen similar results. – Jean Marie Nov 3 '18 at 23:42
• @Jean Marie: Thanks. Actually, the list of related questions was shortened to minimum to make the question more digestible. interestingly, it seems that intuition fails in this case, for example, the number 10 (as well as 11) was completely unexpected. – g.kov Nov 4 '18 at 5:59
• Beautiful, thank you for sharing. There exist souces in French ( chaîne de Steiner or cercle de Steiner). mathafou.free.fr/pbg/sol139.html I like this animation thedudeminds.net/?p=3324 – user376343 Nov 4 '18 at 15:21