{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\\mathbb{R}^2$. Let $M_A$, $M_B$ and $M_C$ be the [[midpoints|0S706JBZBG]] of the sides $BC$, $CA$ and $AB$.\nThese three points are not collinear. The nine-point circle of $ABC$ is the [[circumcircle|03VZJE6C8C]] of the triangle $M_A M_B M_C$.\n\n## Remarks\n\nThe circle also contains the feet of the three [[altitudes|0C9MZY2RNA]], and the midpoints of the segments $AH$, $BH$ and $CH$, where $H$ is the [[orthocentre|0WB619CA0V]]. This gives the nine points of the name.\nIts centre is the midpoint of $H$ and the circumcentre $O$ of $ABC$. Its radius is half of the circumradius of $ABC$.\nBrianchon and Poncelet proved in 1821 that all nine points lie on one circle. In 1822, Feuerbach proved that the circle touches the incircle and the three excircles.\n\n## Explanations\n\n### L1 human\n\nJoin the midpoints of the sides of a triangle to get a smaller triangle. The circle through its corners is the nine-point circle. It also passes through six other special points of the big triangle.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "033H8MQRC3",
    "kind": "definition",
    "msc": [
      "51M04"
    ],
    "name": "Nine-point circle",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Nine-point_circle"
      },
      {
        "locator": "Discovery",
        "url": "https://en.wikipedia.org/wiki/Nine-point_circle"
      },
      {
        "locator": "Other properties of the nine-point circle",
        "url": "https://en.wikipedia.org/wiki/Nine-point_circle"
      }
    ],
    "tag": "0J768YF1ZE",
    "uses": [
      "0CTNFGYMM2",
      "0S706JBZBG",
      "03VZJE6C8C",
      "0C9MZY2RNA",
      "0WB619CA0V"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0J768YF1ZE/",
  "links": [
    {
      "field": "defines",
      "target": "033H8MQRC3"
    },
    {
      "field": "uses",
      "target": "0CTNFGYMM2"
    },
    {
      "field": "uses",
      "target": "0S706JBZBG"
    },
    {
      "field": "uses",
      "target": "03VZJE6C8C"
    },
    {
      "field": "uses",
      "target": "0C9MZY2RNA"
    },
    {
      "field": "uses",
      "target": "0WB619CA0V"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0J768YF1ZE.md",
  "msc": [
    "51M04"
  ],
  "name": "Nine-point circle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0J7/0J768YF1ZE.md",
  "status": null,
  "tag": "0J768YF1ZE",
  "uri": "m2://0J768YF1ZE",
  "url": "https://math2.tamnd.com/wiki/nine-point-circle-definition/",
  "writing": {
    "tag": "0J768YF1ZE",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}