# Difference between revisions of "Euler line"

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− | {{ | + | In any [[triangle]] <math>\triangle ABC</math>, the '''Euler line''' is a [[line]] which passes through the [[orthocenter]] <math>H</math>, [[centroid]] <math>G</math>, [[circumcenter]] <math>O</math>, [[nine-point center]] <math>N</math> and [[De Longchamps point]] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial theorem of Euclidean [[geometry]]. |

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+ | Certain fixed orders and distance [[ratio]]s hold among these points. In particular, <math>\overline{OGNH}</math> and <math>OG:GN:NH = 2:1:3</math> | ||

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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>. | 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>. | ||

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## Revision as of 11:57, 6 July 2007

In any triangle , the **Euler line** is a line which passes through the orthocenter , centroid , circumcenter , nine-point center and De Longchamps point . It is named after Leonhard Euler. Its existence is a non-trivial theorem of Euclidean geometry.

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Certain fixed orders and distance ratios hold among these points. In particular, and

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

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