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]]. | + | 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]]. |
An interesting property of distances between these points on the Euler line: | An interesting property of distances between these points on the Euler line: | ||
* <math>OG:GN:NH\equiv2:1:3</math> | * <math>OG:GN:NH\equiv2:1:3</math> | ||
| − | Construct an [[orthic triangle]]<math>\triangle | + | 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]]. |
Revision as of 11:46, 5 November 2006
This article is a stub. Help us out by expanding it.
Let 
be a triangle, points 
as 
's orthocenter, nine-point center, centroid, circumcenter, De Longchamps point respectively, then these points are collinear(regardless of 's shape). And the line passes through points is called as Euler line, which is named after Leonhard Euler.
An interesting property of distances between these points on the Euler line:
Construct an orthic triangle
, then Euler lines of 
,
,
concur at 's nine-point center.
