{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\Gamma_1, \\Gamma_2, \\Gamma_3, \\Gamma_4$ be four [[circles|060W5CBZP6]] in the Euclidean plane such that each two of them are tangent and the six points of tangency are distinct.\nLet $r_i$ be the radius of $\\Gamma_i$.\nPut $k_i = 1/r_i$ if no other circle lies inside $\\Gamma_i$, and $k_i = -1/r_i$ if the other three circles lie inside $\\Gamma_i$.\nThen\n$$(k_1 + k_2 + k_3 + k_4)^2 = 2\\left(k_1^2 + k_2^2 + k_3^2 + k_4^2\\right).$$\n\n## Remarks\n\nThe numbers $k_i$ are the signed curvatures of the circles.\nThe statement extends to the case where one circle is a straight line, with curvature $0$.\nSolving the quadratic equation for $k_4$ gives\n$$k_4 = k_1 + k_2 + k_3 \\pm 2\\sqrt{k_1 k_2 + k_2 k_3 + k_3 k_1},$$\nand the two signs correspond to the two circles that touch $\\Gamma_1, \\Gamma_2, \\Gamma_3$.\nExample: an outer circle of radius $1$ ($k_1 = -1$) holds two circles of radius $1/2$ ($k_2 = k_3 = 2$).\nThe formula gives $k_4 = 3$, and indeed $(-1 + 2 + 2 + 3)^2 = 36 = 2(1 + 4 + 4 + 9)$.\nOne proof applies an inversion that sends two of the circles to parallel lines and computes directly.\n\n## Explanations\n\n### L1 human\n\nFour circles that all touch each other are tightly linked.\nIf you know the sizes of three of them, the theorem tells you the size of the fourth.\nIt is cleanest when written with curvatures, the reciprocals of the radii.\n\n### L2 human\n\nThe theorem is the basic step in the construction of Apollonian circle packings.\nStarting from four mutually tangent circles, the second root of the quadratic replaces one circle by a new one, and the process repeats.\nIf the first four curvatures are integers, all curvatures in the packing are integers, which makes these packings an object of number theory.\nIt is a special case of the [[problem of Apollonius|F9410EB0N1]] where the given circles touch each other.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "060W5CBZP6"
    ],
    "kind": "statement",
    "msc": [
      "51M04",
      "52C26"
    ],
    "name": "Descartes' theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1196538",
      "wikipedia": "Descartes's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Descartes%27s_theorem"
      }
    ],
    "tag": "J4Y2E05RHQ",
    "uses": [
      "060W5CBZP6"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/J4Y2E05RHQ/",
  "links": [
    {
      "field": "about",
      "target": "060W5CBZP6"
    },
    {
      "field": "uses",
      "target": "060W5CBZP6"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/J4Y2E05RHQ.md",
  "msc": [
    "51M04",
    "52C26"
  ],
  "name": "Descartes' theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J4Y/J4Y2E05RHQ.md",
  "status": null,
  "tag": "J4Y2E05RHQ",
  "uri": "m2://J4Y2E05RHQ",
  "url": "https://tamnd.github.io/math2/wiki/descartes-theorem/",
  "writing": {
    "tag": "J4Y2E05RHQ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}