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 10:57, 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.
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
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 .
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