View source for Sylvester's triangle problem

[[File:Sylvester theorem1.svg|thumb|upright=1.5|sum of three equal lengthed vectors]]
'''Sylvester's theorem''' or '''Sylvester's formula''' describes a particular interpretation of the sum of three pairwise distinct [[vector (mathematics and physics)|vectors]] of equal length in the context of [[triangle geometry]]. It is also referred to as '''Sylvester's (triangle) problem''' in literature, when it is given as a problem rather than a theorem. The theorem is named after the British mathematician [[James Joseph Sylvester]].

== Theorem ==
Consider three pairwise distinct vectors of equal length <math>\vec{u}</math>, <math>\vec{v}</math> and <math>\vec{w}</math> each of them acting on the same point <math>O</math> thus creating the points  <math>A</math>, <math>B</math> and <math>C</math>. Those points form the triangle  <math>\triangle ABC</math> with <math>O</math> as the center of its [[circumcircle]]. Now let <math>H</math> denote the [[orthocenter]] of the triangle, then connection vector  <math>\overrightarrow{OH}</math> is equal to the sum of the three vectors:<ref>Roger A. Johnson: ''Advanced Euclidean Geometry''. Dover 2007, {{ISBN|978-0-486-46237-0}}, p. 251</ref><ref>Heinrich Dörrie: ''100 Great Problems of Elementary Mathematics''. Dover, 1965, {{ISBN|0486-61348-8}}, S. 142 ([https://archive.org/details/100GreatProblemsOfElementaryMathematicsDoverHeinrichDrrie_201808/page/n3/mode/2up online-copy] at the [[internet archive]])</ref>
:<math>\overrightarrow{OH}=\vec{u}+\vec{v}+\vec{w}</math>

Furthermore, since the points <math>O</math> and <math>H</math> are located on the [[Euler line]] together with the [[centroid]] <math>S</math> the following equation holds:<ref name="Villiers">Michael de Villiers: "'Generalising a problem of Sylvester". In: ''The Mathematical Gazette'', volume 96, no. 535 (March 2012), pp 78-81 ([https://www.jstor.org/stable/23249519 JSTOR])</ref>
:<math>\overrightarrow{OH}=3\cdot \overrightarrow{OS}</math>

== Generalisation ==
[[File:Sylvester theorem allgmeiner2.svg|thumb|upright=1.5|sum of three vectors]]
If the condition of equal length in Sylvester's theorem is dropped and one considers merely three arbitrary pairwise distinct vectors, then the equation above does not hold anymore. However, the relation with the centroid remains true, that is:<ref name="Villiers"/>
:<math>3\cdot \overrightarrow{OS}=\vec{u}+\vec{v}+\vec{w}</math>

This follows directly from the [[Centroid#Of a finite set of points|definition of the centroid for a finite set of points in <math>\mathbb{R}^n</math>]], which also yields a version for <math>n</math> vectors acting on <math>O</math>:<ref name="Villiers"/> 
:<math>n\cdot \overrightarrow{OS}=\sum_{i=1}^n v_i</math> 
Here <math>S</math> is the centroid of the vertices of the polygon generated by the <math>n</math> vectors acting on <math>O</math>.<ref>Note that the (area) centroid of a polygon with ''n'' vertices differs from the centroid of its vertices for ''n''>3</ref>

== References ==
<references/>

== External links ==
* {{MathWorld |id=/SylvestersTriangleProblem|title=Sylvester's Triangle Problem}}
*Darij Grinberg: [http://www.cip.ifi.lmu.de/~grinberg/11398.pdf ''Solution to American Mathematical Monthly Problem 11398 by Stanley Huang''] – contains Sylvester's theorem including its proof as a lemma

[[Category:Theorems about triangles]]
[[Category:Triangle problems]]