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[[Image:Triangle.EulerLine.svg|thumb|right|300px|Euler's line (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In [[geometry]], the '''Euler line''', named after [[Leonhard Euler]], is a [[line (mathematics)|line]] determined from any [[triangle]] that is not [[equilateral triangle|equilateral]]; it passes through several important points determined from the triangle. It passes through the [[orthocenter]], the [[circumcenter]], the [[centroid]], the [[Exeter point]] and the center of the [[nine-point circle]] of the triangle.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are [[Line (geometry)|collinear]]. This property is also true for the nine-point center, although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they do not, and the Euler line is determined by any two of them. The center of the nine-point circle lies midway along the Euler line between the orthocenter and the [[circumcenter]], and the distance from the centroid to the circumcenter is half that from the centroid to the orthocenter.
Registered users will be able to choose between the following three rendering modes:


Other notable points that lie on the Euler line are the [[de Longchamps point]], the [[Schiffler point]], the [[Exeter point]] and the [[far-out point]]. However, the [[incenter]] lies on the Euler line only for [[isosceles triangle]]s.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


Let ''A'', ''B'', ''C'' denote the vertex angles of the reference triangle, and let ''x'' : ''y'' : ''z'' be a variable point in [[trilinear coordinates]]; then an equation for the Euler line is
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:<math>\sin 2A \sin(B - C)x + \sin 2B \sin(C - A)y + \sin 2C \sin(A - B)z = 0.\,</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Another particularly useful way to represent the Euler line is in terms of a parameter t.  Starting with the circumcenter (with trilinears <math>\cos A : \cos B : \cos C</math>) and the orthocenter (with trilinears <math>\sec A : \sec B : \sec C = \cos B \cos C : \cos C \cos A : \cos A \cos B)</math>, every point on the Euler line, except the orthocenter, is given as
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


:<math>\cos A + t \cos B \cos C : \cos B + t \cos C \cos A : \cos C + t \cos A \cos B\,</math>
==Demos==


for some ''t''.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Examples:
* [[centroid]] = <math>\cos A + \cos B \cos C : \cos B + \cos C \cos A : \cos C + \cos A \cos B</math>
* [[nine-point center]] = <math>\cos A + 2 \cos B \cos C : \cos B + 2 \cos C \cos A : \cos C + 2 \cos A \cos B</math>
* [[De Longchamps point]] = <math>\cos A - \cos B \cos C : \cos B - \cos C \cos A : \cos C - \cos A \cos B</math>
* [[Euler infinity point]] = <math>\cos A - 2 \cos B \cos C : \cos B - 2 \cos C \cos A : \cos C - 2 \cos A \cos B</math>


==See also==
* accessibility:
*[[Gossard perspector]]
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
*[[Central line (geometry)|Central line]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


== References ==
==Test pages ==


*{{cite journal
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
| author = Euler, Leonhard
*[[Displaystyle]]
| authorlink = Leonhard Euler
*[[MathAxisAlignment]]
| title = Solutio facilis problematum quorundam geometricorum difficillimorum
*[[Styling]]
| journal = Novi Commentarii academiae scientarum imperialis Petropolitanae
*[[Linebreaking]]
| volume = 11
*[[Unique Ids]]
| year = 1767
*[[Help:Formula]]
| pages = 103–123
| url = http://math.dartmouth.edu/~euler/pages/E325.html
| id = <!--Enestrom number-->E325}} Reprinted in ''Opera Omnia'', ser. I, vol. XXVI, pp.&nbsp;139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953, {{MR|0061061}}.


*{{cite journal
*[[Inputtypes|Inputtypes (private Wikis only)]]
| author = Kimberling, Clark
*[[Url2Image|Url2Image (private Wikis only)]]
| title = Triangle centers and central triangles
==Bug reporting==
| journal = Congressus Numerantium
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
| volume = 129
| year = 1998
| pages = i–xxv, 1–295}}
 
== External links ==
* [http://www.cut-the-knot.org/triangle/altEuler.shtml Altitudes and the Euler Line] and [http://www.cut-the-knot.org/triangle/EulerLine.shtml Euler Line and 9-Point Circle] at [[cut-the-knot]]
* [http://faculty.evansville.edu/ck6/tcenters/class/eulerline.html Triangle centers on the Euler line], by Clark Kimberling.
* [http://www.mathopenref.com/eulerline.html An interactive applet showing several triangle centers that lies on the Euler line].
* {{mathworld | title = Euler Line | urlname = EulerLine}}
* [http://demonstrations.wolfram.com/EulerLine/ "Euler Line"] by Eric Rowland, the [[Wolfram Demonstrations Project]], 2007.
* [http://dynamicmathematicslearning.com/ninepointconic.html Nine-point conic and Euler line generalization] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches] Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.
* [http://dynamicmathematicslearning.com/furtherEuler.html A further Euler line generalization] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches] Generalizes the Euler line further by disassociating it from the nine-point conic (see above).
 
[[Category:Triangle geometry]]
 
[[ar:مستقيم أويلر]]
[[bg:Ойлерова права]]
[[ca:Recta d'Euler]]
[[cs:Eulerova přímka]]
[[de:Eulersche Gerade]]
[[es:Recta de Euler]]
[[fr:Droite d'Euler]]
[[ko:오일러선]]
[[it:Retta di Eulero]]
[[he:ישר אוילר]]
[[hu:Euler-egyenes]]
[[nl:Rechte van Euler]]
[[ja:オイラー線]]
[[km:បន្ទាត់អយល័រ]]
[[pms:Reta d'Euler]]
[[pl:Prosta Eulera]]
[[pt:Reta de Euler]]
[[ro:Dreapta lui Euler]]
[[ru:Прямая Эйлера]]
[[sq:Drejtëza e Eulerit]]
[[sl:Eulerjeva premica]]
[[sr:Ојлерова права]]
[[fi:Eulerin suora]]
[[ta:ஆய்லர் கோடு]]
[[uk:Лінія Ейлера]]
[[vi:Đường thẳng Euler]]
[[zh:歐拉線]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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