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?