# Difference between revisions of "Euler line"

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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>. | ||

− | ==Proof | + | ==Proof Centroid Lies on Euler Line== |

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. | 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. | ||

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

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

+ | |||

+ | ==Proof Nine-Point Center Lies on Euler Line== | ||

+ | Assuming that the [[nine point circle]] exists and that <math>N</math> is the center, note that a homothety centered at <math>H</math> with factor <math>2</math> brings the [[Euler point]]s <math>{E_A, E_B, E_C\}</math> onto the circumcircle of <math>\triangle ABC</math>. Thus, it brings the nine-point circle to the circumcircle. Additionally, <math>N</math> should be sent to <math>O</math>, thus <math>N \in \overline{HO}</math> and <math>\frac{HN}{ON} = 1</math>. | ||

~always_correct | ~always_correct |

## Revision as of 19:35, 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 Centroid Lies on Euler Line

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 .

## Proof Nine-Point Center Lies on Euler Line

Assuming that the nine point circle exists and that is the center, note that a homothety centered at with factor brings the Euler points ${E_A, E_B, E_C\}$ (Error compiling LaTeX. ! Missing } inserted.) onto the circumcircle of . Thus, it brings the nine-point circle to the circumcircle. Additionally, should be sent to , thus and .

~always_correct

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