{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\\mathbb{R}^2$. For [[collinear points|0AFNNTRTJE]] $X$, $Y$, $Z$ with $Z \\neq Y$, let the signed ratio $\\frac{XZ}{ZY}$ be the real number $\\lambda$ with $Z - X = \\lambda\\,(Y - Z)$.\nLet $O$ be a point that is not on the [[lines|0MP9BCF6DP]] $BC$, $CA$ and $AB$. Suppose that the lines $AO$, $BO$ and $CO$ meet the lines $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$.\nThen\n$$\\frac{AF}{FB} \\cdot \\frac{BD}{DC} \\cdot \\frac{CE}{EA} = 1.$$\nConversely, let $D$, $E$ and $F$ be points of the lines $BC$, $CA$ and $AB$, different from the vertices, with this product equal to $1$. Then the lines $AD$, $BE$ and $CF$ meet in one point or are all parallel.\n\n## Remarks\n\nThe signed ratio $\\frac{AF}{FB}$ is positive if $F$ is between $A$ and $B$, and negative otherwise.\nThe theorem uses only ratios of segments on one line. So it is true in each affine plane over each field.\nFor the three medians, each ratio is $1$. The converse then shows again that the medians meet in one point.\n\n## History\n\nGiovanni Ceva published the theorem in 1678, in his work De lineis rectis. Yusuf al-Mu'taman ibn Hud, an 11th century king of Zaragoza, had proved it much earlier.\n\n## Explanations\n\n### L1 human\n\nDraw three lines from the corners of a triangle through one common inside point. Each line cuts the opposite side into two parts. Multiply the three ratios of the parts, going round the triangle in one direction: the product is always $1$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "51M04"
    ],
    "name": "Ceva's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q213603",
      "wikipedia": "Ceva's theorem"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Ceva%27s_theorem"
      }
    ],
    "tag": "0KN93VDSXA",
    "uses": [
      "0CTNFGYMM2",
      "0MP9BCF6DP",
      "0AFNNTRTJE"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0KN93VDSXA/",
  "links": [
    {
      "field": "uses",
      "target": "0CTNFGYMM2"
    },
    {
      "field": "uses",
      "target": "0MP9BCF6DP"
    },
    {
      "field": "uses",
      "target": "0AFNNTRTJE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0KN93VDSXA.md",
  "msc": [
    "51M04"
  ],
  "name": "Ceva's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0KN/0KN93VDSXA.md",
  "status": null,
  "tag": "0KN93VDSXA",
  "uri": "m2://0KN93VDSXA",
  "url": "https://math2.tamnd.com/wiki/cevas-theorem/",
  "writing": {
    "tag": "0KN93VDSXA",
    "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
      }
    ]
  }
}