# 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 fact of Euclidean [[geometry]]. | + | 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]]. 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> |

− | [[ | + | 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>. |

− | + | ==Proof of Existence== | |

+ | This proof utilizes the concept of [[spiral similarity]], which in this case is a [[rotation]] followed [[homothety]]. Consider the [[medial triangle]] <math>\triangle O_AO_BO_C</math>. It is similar to <math>\triangle ABC</math>. Specifically, a rotation of <math>180^\circ</math> about the midpoint of <math>O_BO_C</math> followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us examine what else this transformation, which we denote as <math>\mathcal{S}</math>, will do. | ||

+ | It turns out <math>O</math> is the orthocenter, and <math>G</math> is the centroid of <math>\triangle O_AO_BO_C</math>. Thus, <math>\mathcal{S}(\{O_A, O, G\}) = \{A, H, G\}</math>. As a homothety preserves angles, it follows that <math>\measuredangle O_AOG = \measuredangle AHG</math>. Finally, as <math>\overline{AH} || \overline{O_AO}</math> it follows that | ||

+ | <cmath>\triangle AHG = \triangle O_AOG</cmath> | ||

+ | Thus, <math>O, G, H</math> are collinear, and <math>\frac{OG}{HG} = \frac{1}{2}</math>. | ||

+ | |||

+ | ~always_correct | ||

+ | |||

+ | |||

+ | [[Image:Euler Line.PNG||500px|frame|center]] | ||

− | |||

{{stub}} | {{stub}} |

## Revision as of 15:15, 3 August 2017

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 fact of Euclidean geometry. 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 .

## Proof of Existence

This proof utilizes the concept of spiral similarity, which in this case is a rotation followed homothety. Consider the medial triangle . It is similar to . Specifically, a rotation of about the midpoint of followed by a homothety with scale factor centered at brings . Let us examine what else this transformation, which we denote as , will do.

It turns out is the orthocenter, and is the centroid of . Thus, . As a homothety preserves angles, it follows that . Finally, as it follows that Thus, are collinear, and .

~always_correct

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