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{{dablink|For another use of the term '''median''' in geometry, see [[Geometric median]].}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
[[Image:Triangle.Centroid.svg|right|thumb|The triangle medians and the centroid.]]
In [[geometry]], a '''median''' of a [[triangle]] is a [[line segment]] joining a [[vertex (geometry)|vertex]] to the [[midpoint]] of the opposing side. Every triangle has exactly three medians: one running from each vertex to the opposite side. In the case of [[isosceles]] and [[equilateral]] triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.


==Relation to center of mass==
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]]
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* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


Each median of a triangle passes through the triangle's [[centroid]], which is the center of mass of an object of uniform density in the shape of the triangle. Thus the object would balance on any line through the centroid, including any median.
Registered users will be able to choose between the following three rendering modes:


==Equal-area division==
'''MathML'''
[[Image:Triangle.Centroid.Median.png|thumb|right|300px|]]
:<math forcemathmode="mathml">E=mc^2</math>
Each median divides the area of the triangle in half; hence the name. (Any other lines which divide the area of the triangle into two equal parts do not pass through the centroid.)[http://www.btinternet.com/~se16/js/halfarea.htm] The three medians divide the triangle into six smaller triangles of equal [[area]].
===Proof===
Consider a triangle ''ABC'' Let ''D'' be the midpoint of <math>\overline{AB}</math>, ''E'' be the midpoint of <math>\overline{BC}</math>, ''F'' be the midpoint of <math>\overline{AC}</math>, and ''O'' be the centroid.


By definition, <math>AD=DB, AF=FC, BE=EC \,</math>. Thus <math>[ADO]=[BDO], [AFO]=[CFO], [BEO]=[CEO],</math> and <math>[ABE]=[ACE] \,</math>, where <math>[ABC]</math> represents the [[area]] of triangle <math>\triangle ABC</math> ; these hold because in each case the two triangles have bases of equal length and share a common altitude from the (extended) base, and a triangle's area equals one-half its base times its height.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


We have:
'''source'''
:<math>[ABO]=[ABE]-[BEO] \,</math>
:<math forcemathmode="source">E=mc^2</math> -->


:<math>[ACO]=[ACE]-[CEO] \,</math>
<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].


Thus, <math>[ABO]=[ACO] \,</math> and <math>[ADO]=[DBO], [ADO]=\frac{1}{2}[ABO]</math>
==Demos==


Since <math>[AFO]=[FCO], [AFO]= \frac{1}{2}ACO=\frac{1}{2}[ABO]=[ADO]</math>, therefore, <math>[AFO]=[FCO]=[DBO]=[ADO]\,</math>.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
Using the same method, you can show that <math>[AFO]=[FCO]=[DBO]=[ADO]=[BEO]=[CEO] \,</math>.


==Formulas involving the medians' lengths==
The lengths of the medians can be obtained from [[Apollonius' theorem]] as:


:<math>m_a = \sqrt {\frac{2 b^2 + 2 c^2 - a^2}{4} }, </math>
* accessibility:
** 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]]
** [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.


:<math>m_b = \sqrt {\frac{2 a^2 + 2 c^2 - b^2}{4} }, </math>
==Test pages ==


:<math>m_c = \sqrt {\frac{2 a^2 + 2 b^2 - c^2}{4} }, </math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


where ''a'', ''b'' and ''c'' are the sides of the triangle with respective medians ''m''<sub>''a''</sub>, ''m''<sub>''b''</sub>, and ''m''<sub>''c''</sub> from their midpoints.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
Thus we have the relationships:<ref>{{cite book |last=Déplanche |first=Y. |title=Diccio fórmulas  |language= |others=Medianas de un triángulo |year=1996 |publisher=Edunsa |isbn=978-84-7747-119-6 |page=22 |url=http://books.google.com/books?id=1HVHOwAACAAJ |accessdate=2011-04-24 }}</ref>
==Bug reporting==
 
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 .
:<math>a = \frac{2}{3} \sqrt{-m_a^2 + 2m_b^2 + 2m_c^2} = \sqrt{2(b^2+c^2)-4m_a^2} = \sqrt{\frac{b^2}{2} - c^2 + 2m_b^2} = \sqrt{\frac{c^2}{2} - b^2 + 2m_c^2},</math>
 
:<math>b = \frac{2}{3} \sqrt{-m_b^2 + 2m_a^2 + 2m_c^2} = \sqrt{2(a^2+c^2)-4m_b^2} = \sqrt{\frac{a^2}{2} - c^2 + 2m_a^2} = \sqrt{\frac{c^2}{2} - a^2 + 2m_c^2},</math>
 
:<math>c = \frac{2}{3} \sqrt{-m_c^2 + 2m_b^2 + 2m_a^2} = \sqrt{2(b^2+a^2)-4m_c^2} = \sqrt{\frac{b^2}{2} - a^2 + 2m_b^2} = \sqrt{\frac{a^2}{2} - b^2 + 2m_a^2}.</math>
 
==Other properties==
 
The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex.
 
For any triangle,<ref name=P&S>Posamentier, Alfred S., and Salkind, Charles T., ''Challenging Problems in Geometry'', Dover, 1996: pp. 86-87.</ref> 
 
:<math>\tfrac{3}{4}</math>(perimeter) < sum of the medians < <math>\tfrac{3}{2}</math>(perimeter).
 
For any triangle with sides <math>a, b, c</math> and medians <math>m_a, m_b, m_c</math>,<ref name=P&S/>
 
:<math>\tfrac{3}{4}(a^2+b^2+c^2)=m_a^2+m_b^2+m_c^2.</math>
 
==See also==
*[[Angle bisector]]
*[[Altitude (triangle)]]
 
==References==
 
{{reflist}}
 
==External links==
{{Commons cat|Median (geometry)}}
* [http://www.btinternet.com/~se16/js/halfarea.htm Medians and Area Bisectors of a Triangle]
* [http://www.cut-the-knot.org/triangle/medians.shtml The Medians] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/MedianTriangle.shtml Area of Median Triangle] at [[cut-the-knot]]
* [http://www.mathopenref.com/trianglemedians.html Medians of a triangle] With interactive animation
* [http://www.mathopenref.com/constmedian.html Constructing a median of a triangle with compass and straightedge] animated demonstration
* {{MathWorld |title=Triangle Median |urlname=TriangleMedian}}
 
[[Category:Elementary geometry]]
[[Category:Triangles]]
[[Category:Articles containing proofs]]
 
[[ar:متوسط (هندسة رياضية)]]
[[ast:Mediana (xeometría)]]
[[zh-min-nan:Tiong-soàⁿ (kí-hô-ha̍k)]]
[[bg:Медиана]]
[[ca:Mitjana (geometria)]]
[[cs:Těžnice]]
[[de:Seitenhalbierende]]
[[et:Mediaan (geomeetria)]]
[[el:Διάμεσος (γεωμετρία)]]
[[es:Mediana (geometría)]]
[[eo:Mediano (geometrio)]]
[[fa:میانه مثلث]]
[[fr:Médiane (géométrie)]]
[[gl:Mediana (xeometría)]]
[[ko:중선]]
[[it:Mediana (geometria)]]
[[he:תיכון (גאומטריה)]]
[[lv:Mediāna]]
[[hu:Súlyvonal]]
[[nl:Zwaartelijn]]
[[ja:中線]]
[[km:មេដ្យាន]]
[[pl:Środkowa trójkąta]]
[[pt:Mediana (geometria)]]
[[ro:Mediană]]
[[ru:Медиана треугольника]]
[[sl:Mediana (geometrija)]]
[[ta:நடுக்கோடு (வடிவவியல்)]]
[[th:เส้นมัธยฐาน]]
[[tr:Kenarortay]]
[[uk:Медіана]]
[[zh:中线]]

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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