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 13:53, 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 .
