{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $A$, $B$, $C$ and $D$ be [[concyclic points|049DCJZ448]] of the Euclidean plane $\\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Suppose that the segments $AC$ and $BD$ meet in a point that is inside both segments. Thus $ABCD$ is a convex [[cyclic quadrilateral|0RCP929Y1M]] with the diagonals $AC$ and $BD$.\nThen\n$$d(A, C)\\, d(B, D) = d(A, B)\\, d(C, D) + d(B, C)\\, d(A, D).$$\n\n## Remarks\n\nIn words: the product of the diagonals is the sum of the products of the opposite sides.\nThe converse is also true. If a quadrilateral has this property, then it is cyclic.\nFor each four points of the plane, $d(A, C)\\, d(B, D) \\leq d(A, B)\\, d(C, D) + d(B, C)\\, d(A, D)$. This is Ptolemy's inequality.\nFor a rectangle with the sides $a$ and $b$, the theorem gives $a^2 + b^2 = c^2$, with the diagonal $c$. This is the [[Pythagorean theorem|0VP3A1290K]].\n\n## History\n\nThe theorem has the name of the astronomer Ptolemy. He used it to make his table of chords, a trigonometric table for astronomy.\n\n## Explanations\n\n### L1 human\n\nPut four points in order on a circle and join them into a quadrilateral. Multiply the two diagonals. You get the same number as when you multiply the opposite sides in pairs and add the two products.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "51M04"
    ],
    "name": "Ptolemy's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q459547",
      "wikipedia": "Ptolemy's theorem"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Ptolemy%27s_theorem"
      },
      {
        "locator": "Ptolemy's inequality",
        "url": "https://en.wikipedia.org/wiki/Ptolemy%27s_theorem"
      }
    ],
    "tag": "058YS05YP5",
    "uses": [
      "0RCP929Y1M",
      "049DCJZ448",
      "0NPPGHQNXQ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/058YS05YP5/",
  "links": [
    {
      "field": "uses",
      "target": "0RCP929Y1M"
    },
    {
      "field": "uses",
      "target": "049DCJZ448"
    },
    {
      "field": "uses",
      "target": "0NPPGHQNXQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/058YS05YP5.md",
  "msc": [
    "51M04"
  ],
  "name": "Ptolemy's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/058/058YS05YP5.md",
  "status": null,
  "tag": "058YS05YP5",
  "uri": "m2://058YS05YP5",
  "url": "https://math2.tamnd.com/wiki/ptolemys-theorem/",
  "writing": {
    "tag": "058YS05YP5",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}