Difference between revisions of "Euler line"

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Let <math>ABC</math> be a triangle, points <math>H, N, G, O, L</math> as <math>\triangle ABC</math>'s [[orthocenter]], [[nine-point center]], [[centroid]], [[circumcenter]], [[De Longchamps point]] respectively, then these points are [[collinear]](regardless of <math>\triangle ABC</math>'s shape). And the line passes through points <math>H, N, G, O, L</math> is called as Euler line, which is named after [[Leonhard Euler]].
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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]].
  
An interesting property of distances between these points on the Euler line:
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Certain fixed [[ratio]]s hold among the distances between these points:
* <math>OG:GN:NH\equiv2:1:3</math>
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* <math>OG:GN:NH = 2:1:3</math>
  
Construct an [[orthic triangle]]<math>\triangle H_AH_BH_C</math>, then Euler lines of <math>\triangle AH_BH_C</math>,<math>\triangle BH_CH_A</math>,<math>\triangle CH_AH_B</math> concur at <math>\triangle ABC</math>'s [[nine-point center]].
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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>.

Revision as of 14:53, 5 November 2006

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

Let $\triangle ABC$ be a triangle with orthocenter $H$, nine-point center $N$, centroid $G$, circumcenter $O$ and De Longchamps point $L$. Then these points are collinear and the line passes through points $H, N, G, O, L$ is called the Euler line of $\triangle ABC$. It is named after Leonhard Euler.

Certain fixed ratios hold among the distances between these points:

  • $OG:GN:NH = 2:1:3$

Given the orthic triangle$\triangle H_AH_BH_C$ of $\triangle ABC$, the Euler lines of $\triangle AH_BH_C$,$\triangle BH_CH_A$, and $\triangle CH_AH_B$ concur at $N$, the nine-point center of $\triangle ABC$.

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