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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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 fact of Euclidean [[geometry]].
  
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[[Image:Euler Line.PNG|500px|thumb|The Euler Line|right]]
  
 
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>
 
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>

Revision as of 22:49, 23 September 2007

In any triangle $\triangle ABC$, the Euler line is a line which passes through the orthocenter $H$, centroid $G$, circumcenter $O$, nine-point center $N$ and De Longchamps point $L$. It is named after Leonhard Euler. Its existence is a non-trivial fact of Euclidean geometry.

The Euler Line

Certain fixed orders and distance ratios hold among these points. In particular, $\overline{OGNH}$ and $OG:GN:NH = 2:1:3$


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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