# Difference between revisions of "Euler line"

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− | Let <math>ABC</math> be a triangle | + | Let <math>\triangle ABC</math> be a [[triangle]] with [[orthocenter]] <math>H</math>, [[nine-point center]] <math>N</math>, [[centroid]] <math>G</math>, [[circumcenter]] <math>O</math> and [[De Longchamps point]] <math>L</math>. Then these points are [[collinear]] and the line passes through points <math>H, N, G, O, L</math> is called the '''Euler line''' of <math>\triangle ABC</math>. It is named after [[Leonhard Euler]]. |

− | + | Certain fixed [[ratio]]s hold among the distances between these points: | |

− | * <math>OG:GN:NH | + | * <math>OG:GN:NH = 2:1:3</math> |

− | + | Given the [[orthic triangle]]<math>\triangle H_AH_BH_C</math> of <math>\triangle ABC</math>, the Euler lines of <math>\triangle AH_BH_C</math>,<math>\triangle BH_CH_A</math>, and <math>\triangle CH_AH_B</math> [[concurrence | concur]] at <math>N</math>, the nine-point center of <math>\triangle ABC</math>. |

## Revision as of 14:53, 5 November 2006

*This article is a stub. Help us out by expanding it.*

Let be a triangle with orthocenter , nine-point center , centroid , circumcenter and De Longchamps point . Then these points are collinear and the line passes through points is called the **Euler line** of . It is named after Leonhard Euler.

Certain fixed ratios hold among the distances between these points:

Given the orthic triangle of , the Euler lines of ,, and concur at , the nine-point center of .