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  "body": "\n## Example\n\nLet $\\alpha$ and $\\beta$ be two [[circles|060W5CBZP6]] in the plane with $\\beta$ inside $\\alpha$ and no common point.\nA *Steiner chain* for $\\alpha$ and $\\beta$ is a cyclic sequence of $n \\ge 3$ circles $\\gamma_1, \\ldots, \\gamma_n$ with two properties.\nEach $\\gamma_i$ is tangent to $\\alpha$ from inside and to $\\beta$ from outside.\nEach $\\gamma_i$ is tangent to $\\gamma_{i+1}$, where $\\gamma_{n+1} = \\gamma_1$.\n\nTake $\\alpha$ of radius $3$ and $\\beta$ of radius $1$, both centred at the origin.\nLet $\\gamma_k$ be the circle of radius $1$ centred at $2(\\cos(k\\pi/3), \\sin(k\\pi/3))$ for $k = 1, \\ldots, 6$.\nThe centre of $\\gamma_k$ is at distance $2 = 3 - 1$ from the origin, so $\\gamma_k$ touches $\\alpha$ from inside.\nIt is also at distance $2 = 1 + 1$, so $\\gamma_k$ touches $\\beta$ from outside.\nTwo 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.\nSo these six circles form a Steiner chain.\n\n## Remarks\n\nFor concentric circles of radii $R \u003e r$, a chain of $n$ circles that goes once around $\\beta$ exists exactly when $\\sin(\\pi/n) = (R - r)/(R + r)$.\nSteiner's porism says that if one chain closes up for $\\alpha$ and $\\beta$, then a chain closes up starting from any first circle tangent to both.\nThe proof uses an [[inversion in a circle|YXST7NGMTC]] that sends $\\alpha$ and $\\beta$ to concentric circles, where rotation moves one chain to any other.\n\n## Explanations\n\n### L1 human\n\nPut a small coin inside a large ring, and fill the space between them with coins that touch both and touch their neighbours.\nFor the right sizes, the coins close up into a full necklace.\nThen the necklace can be rolled around: start it from any position and it still closes up.\n",
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