Steiner chain

Example

Let $\alpha$ and $\beta$ be two circles in the plane with $\beta$ inside $\alpha$ and no common point. A Steiner chain for $\alpha$ and $\beta$ is a cyclic sequence of $n \ge 3$ circles $\gamma_1, \ldots, \gamma_n$ with two properties. Each $\gamma_i$ is tangent to $\alpha$ from inside and to $\beta$ from outside. Each $\gamma_i$ is tangent to $\gamma_{i+1}$, where $\gamma_{n+1} = \gamma_1$.

Take $\alpha$ of radius $3$ and $\beta$ of radius $1$, both centred at the origin. Let $\gamma_k$ be the circle of radius $1$ centred at $2(\cos(k\pi/3), \sin(k\pi/3))$ for $k = 1, \ldots, 6$. The centre of $\gamma_k$ is at distance $2 = 3 - 1$ from the origin, so $\gamma_k$ touches $\alpha$ from inside. It is also at distance $2 = 1 + 1$, so $\gamma_k$ touches $\beta$ from outside. Two consecutive centres are at distance $2 \cdot 2 \sin(\pi/6) = 2$, the sum of the radii, so $\gamma_k$ and $\gamma_{k+1}$ touch. So these six circles form a Steiner chain.