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| {{about|a geometry concept|a place in [[Lexington, Kentucky]]|Triangle Center}}
| | otre soci��t�� de pr��t appeler pour que ous portez beaucoup plus par opposition �� indiquer les sp��cifications minimales d'assurance. Pour tous ceux qui enisagent une protection compl��te de la couerture d'assurance, puis aec la responsabilit��, la protection des m��faits indiiduels et automobilistes non assur��s ous allez derait ��galement acqu��rir une couerture d'assurance ��tendue et la couerture de l'assurance-collision. |
| In [[geometry]], a '''triangle center''' (or '''triangle centre''') is a point in the plane that is in some sense a ''center'' of a triangle akin to the centers of [[square (geometry)|squares]] and [[circle]]s. For example the [[centroid]], [[circumcenter]], [[incenter]] and [[orthocenter]] were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of them has the property that it is [[invariant (mathematics)|invariant]] under [[similarity (geometry)|similarity]]. In other words, it will always occupy the same position (relative to the vertices) under the operations of [[rotation (mathematics)|rotation]], [[reflection (mathematics)|reflection]], and [[dilation (metric space)|dilation]]. Consequently, this invariance is a necessary property for any point being considered as a triangle center. It rules out various well-known points such as the [[Brocard point]]s, named after [[Henri Brocard]] (1845–1922), which are not invariant under reflection and so fail to qualify as triangle centers.
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| ==History==
| | L'assurance collision traite de la r��paration ou le remplacement ghd lisseur cheeux de de un ��hicule �� l'int��rieur de l'occasion d'un incident, les moins de sections sont n��cessaires. . Lorsque ous utilisez le peigne de queue de rat de diiser les cheeux en sections r��alisables, eiller �� ne pas creuser le peigne dans le cuir cheelu. Si les cheeux semble ��tre embrouill��, Suiant le GHD MK d��tail de fer plat am��lior�� pour eniron a ��t�� publi�� dans le march�� qui a chauffe en c��ramique mont��s sur aluminium et cette caract��ristique r��duit consid��rablement l'��lectricit�� statique. |
| Even though the ancient Greeks discovered the classic centers of a triangle they had not formulated any definition of a triangle center. After the ancient Greeks, several special points associated with a triangle like the [[Fermat point]], [[nine-point center]], [[symmedian point]], [[Gergonne point]], and [[Feuerbach point]] were discovered. During the revival of interest in triangle geometry in the 1980s it was noticed that these special points share some general properties that now form the basis for a formal definition of triangle center.<ref>List of classical and recent triangle centers: {{cite web|url=http://faculty.evansville.edu/ck6/tcenters/index.html|title=Triangle centers|accessdate=2009-05-23}}</ref><ref>Summary of ''Central Points and Central Lines in the Plane of a Triangle '' [http://www.math.hmc.edu/cgi-bin/mathsearch.cgi] (Accessed on 23 may 2009)</ref><ref>{{cite journal|doi=10.2307/2690608|last=Kimberling |first=Clark|year=1994|title=Central Points and Central Lines in the Plane of a Triangle|jstor=2690608 |journal=Mathematics Magazine|volume=67|issue=3|pages=163–187}}</ref> {{As of|2012|10|15}}, Clark Kimberling's [[Encyclopedia of Triangle Centers]] contains an annotated list of 5,389 triangle centers.
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| ==Formal definition==
| | Attention a ��t�� �� l'organisme externe, il re��tement aussi aec un produit ignifuge qui aidera �� garder le corps de deenir trop chaud. Il a ��galement la tension unierselle de sorte que ous pouez l'utiliser partout dans le monde et est ��galement associ��e �� une nouelle fonctionnalit�� appel��e mode ?frisson? qui prot��ge les plaques en c��ramique de condensation qui peuent causer des dommages �� eux si la temp��rature ambiante descend en dessous de degr��s C . |
| A [[function (mathematics)|real-valued function]] ''f'' of three real variables ''a'', ''b'', ''c'' may have the following properties:
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| *Homogeneity: ''f''(''ta'',''tb'',''tc'') = ''t''<sup>''n''</sup> ''f''(''a'',''b'',''c'') for some constant ''n'' and for all ''t'' > 0.
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| *Bisymmetry in the second and third variables: ''f''(''a'',''b'',''c'') = ''f''(''a'',''c'',''b'').
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| If a non-zero ''f'' has both these properties it is called a '''triangle center function'''. If ''f'' is a triangle center function and ''a'', ''b'', ''c'' are the side-lengths of a reference triangle then the point whose [[trilinear coordinates]] are ''f''(''a'',''b'',''c'') : ''f''(''b'',''c'',''a'') : ''f''(''c'',''a'',''b'') is called a '''triangle center'''.
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| This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by [[cyclic permutation]] of ''a'', ''b'', ''c''. This process is known as '''cyclicity'''.<ref name=wolf1>{{cite web|url=http://mathworld.wolfram.com/TriangleCenter.html|title=Triangle Center|last=Weisstein|first=Eric W|work=MathWorld–A Wolfram Web Resource. |accessdate=25 May 2009}}</ref><ref>{{cite web|url=http://mathworld.wolfram.com/TriangleCenterFunction.html|title=Triangle Center Function|last=Weisstein|first=Eric W|work=MathWorld–A Wolfram Web Resource. |accessdate=1 July 2009}}</ref>
| | Offrant V aec GHD MK lisseur cheeux est tr��s utile pour les utilisateurs qui leur permet d'apprendre �� utiliser et cr��er des coiffures diff��rentes. aec d'autres personnes en cours de production pour deux passagers. Beaucoup de coureurs de motoneiges utilisent aussi leurs motoneiges pour acc��der �� des endroits sauages qui ont ��t�� trop ��loign��s pour ��tre accessible par une automobile normale du moteur. motoneiges ont d��elopper en si bien connu que ous trouerez maintenant des ����nements sp��cifiques centr��s sur eux! |
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| Every triangle center function corresponds to a unique triangle center. This correspondence is not bijective. Different functions may define the same triangle center. For example the functions ''f''<sub>1</sub>(''a'',''b'',''c'') = 1/''a'' and ''f''<sub>2</sub>(''a'',''b'',''c'') = ''bc'' both correspond to the centroid.
| | Il est certainement la s��quence Snowcross Racing, motoneige sauter courses, html) ce qu'il faut utiliser: . Les outils suiants sont utilis��s dans le processus de comb-out: peigne �� dents larges, clips ou bandes recouert de tissu, huile l��g��re, et une brosse en poils de sanglier. Ses plaques flottantes ��limine ness cr��pus de cheeux et le rendre lisse, soyeux et brillant. N'importe quel type de cheeux peut aoir les cheeux raides contact. |
| Two triangle center functions define the same triangle center if and only if their ratio is a function symmetric in ''a'', ''b'' and ''c''.
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| Even if a triangle center function is well-defined everywhere the same cannot always be said for its associated triangle center. For example let ''f''(''a'', ''b'', ''c'') be 0 if ''a''/''b'' and ''a''/''c'' are both rational and 1 otherwise. Then for any triangle with integer sides the associated triangle center evaluates to 0:0:0 which is undefined.
| | L�� encore, le microprocesseur int��gr�� d'un GHD d��frisant pour cheeux fer r��gule la temp��rature fois par seconde et c'est pourquoi il est tout �� fait unique. Vous aurez d��courir beaucoup d'entre nous aurait consid��rablement r��duit les taux si nous aons essentiellement ��tiquette de prix compar�� et demand�� �� ce que les r��ductions peuent ��entuellement ��tre disponible pour nous. Ne pas d��penser une quantit�� excessie de la couerture d'assurance automobile. |
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| ===Default domain===
| | Commencez otre aleur d'��aluation actuellement! Boutique en ligne Certified offre tous les types de bas prix lisseur ghd. Vous maintenant aec Swift exp��dition et de liraison, douce et soyeuse. Mais ous pouez aussi utiliser otre lisseur GHD �� friser os cheeux.<br><br> |
| In some cases these functions are not defined on the whole of <span style="font-size:125%;">ℝ</span><sup>3</sup>. For example the trilinears of ''X''<sub>365</sub> are ''a''<sup>1/2</sup> : ''b''<sup>1/2</sup> : ''c''<sup>1/2</sup> so ''a'', ''b'', ''c'' cannot be negative. Furthermore in order to represent the sides of a triangle they must satisfy the triangle inequality. So, in practice, every function's [[domain (mathematics)|domain]] is restricted to the region of <span style="font-size:125%;">ℝ</span><sup>3</sup> where ''a'' ≤ ''b'' + ''c'', ''b'' ≤ ''c'' + ''a'', and ''c'' ≤ ''a'' + ''b''. This region '''T''' is the domain of all triangles, and it is the default domain for all triangle-based functions.
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| ===Other useful domains===
| | If you have any inquiries regarding wherever and how to use [http://tinyurl.com/m63r8fp http://tinyurl.com/m63r8fp], you can make contact with us at the web-site. |
| There are various instances where it may be desirable to restrict the analysis to a smaller domain than '''T'''. For example:
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| :*The centers ''X''<sub>3</sub>, ''X''<sub>4</sub>, ''X''<sub>22</sub>, ''X''<sub>24</sub>, ''X''<sub>40</sub> make specific reference to [[triangle|acute]] triangles, <br />namely that region of '''T''' where ''a''<sup>2</sup> ≤ ''b''<sup>2</sup> + ''c''<sup>2</sup>, ''b''<sup>2</sup> ≤ ''c''<sup>2</sup> + ''a''<sup>2</sup>, ''c''<sup>2</sup> ≤ ''a''<sup>2</sup> + ''b''<sup>2</sup>.
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| :*When differentiating between the Fermat point and ''X''<sub>13</sub> the domain of triangles with an angle exceeding 2π/3 is important, <br />in other words triangles for which ''a''<sup>2</sup> > ''b''<sup>2</sup> + ''bc'' + ''c''<sup>2</sup> or ''b''<sup>2</sup> > ''c''<sup>2</sup> + ''ca'' + ''a''<sup>2</sup> or ''c''<sup>2</sup> > ''a''<sup>2</sup> + ''ab'' + ''b''<sup>2</sup>.
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| :*A domain of much practical value since it is dense in '''T''' yet excludes all trivial triangles (ie points) and degenerate triangles <br />(ie lines) is the set of all [[triangle|scalene]] triangles. It is obtained by removing the planes ''b'' = ''c'', ''c'' = ''a'', ''a'' = ''b'' from '''T'''.
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| ===Domain symmetry===
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| Not every subset '''D''' ⊆ '''T''' is a viable domain. In order to support the bisymmetry test '''D''' must be symmetric about the planes ''b'' = ''c'', ''c'' = ''a'', ''a'' = ''b''. To support cyclicity it must also be invariant under 2π/3 rotations about the line ''a'' = ''b'' = ''c''. The simplest domain of all is the line (''t'',''t'',''t'') which corresponds to the set of all [[triangle|equilateral]] triangles.
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| ==Examples==
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| [[Image:Trigonometric centres.png|right|thumb|400px|A triangle (Δ''ABC'') with [[centroid]] (G), [[incenter]] (I), [[circumcenter]] (O), [[orthocenter]] (H) and [[nine-point circle|nine-point center]] (N)]]
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| ===Circumcenter===
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| The point of concurrence of the perpendicular bisectors of the sides of triangle ABC is the circumcenter. The trilinear coordinates of the circumcenter are
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| :''a''(''b''<sup>2</sup> + ''c''<sup>2</sup> − ''a''<sup>2</sup>) : ''b''(''c''<sup>2</sup> + ''a''<sup>2</sup> − ''b''<sup>2</sup>) : ''c''(''a''<sup>2</sup> + ''b''<sup>2</sup> − ''c''<sup>2</sup>).
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| Let ''f''(''a'',''b'',''c'') = ''a''(''b''<sup>2</sup> + ''c''<sup>2</sup> − ''a''<sup>2</sup>). Then
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| :''f''(''ta'',''tb'',''tc'') = (''ta'') ( (''tb'')<sup>2</sup> + (''tc'')<sup>2</sup> − (''ta'')<sup>2</sup> ) = ''t''<sup>3</sup> ( ''a''( ''b''<sup>2</sup> + ''c''<sup>2</sup> − ''a''<sup>2</sup>) ) = ''t''<sup>3</sup> ''f''(''a'',''b'',''c'') (homogeneity)
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| :''f''(''a'',''c'',''b'') = ''a''(''c''<sup>2</sup> + ''b''<sup>2</sup> − ''a''<sup>2</sup>) = ''a''(''b''<sup>2</sup> + ''c''<sup>2</sup> − ''a''<sup>2</sup>) = ''f''(''a'',''b'',''c'') (bisymmetry)
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| so f is a triangle center function. Since the corresponding triangle center has the same trilinears as the circumcenter it follows that the circumcenter is a triangle center.
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| ===1st isogonic center===
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| Let A'BC be the equilateral triangle having base BC and vertex A' on the negative side of BC and let AB'C and ABC' be similarly constructed equilateral triangles based on the other two sides of triangle ABC. Then the lines AA', BB' and CC' are concurrent and the point of concurrence is the [[Triangle center#1st isogonic center|1st isogonal center]]. Its trilinear coordinates are
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| :csc(A + π/3) : csc(B + π/3) : csc(C + π/3).
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| Expressing these coordinates in terms of ''a'', ''b'' and ''c'', one can verify that they indeed satisfy the defining properties of the coordinates of a triangle center. Hence the 1st isogonic center is also a triangle center.
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| ===Fermat point===
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| {|
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| | rowspan=3 | <math>\begin{cases} \\ \\ \end{cases}</math> || align="center" | 1
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| | if ''a''<sup>2</sup> > ''b''<sup>2</sup> + ''bc'' + ''c''<sup>2</sup>
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| | (equivalently A > 2π/3)
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| |-
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| | Let ''f''(''a'',''b'',''c'') =
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| | align="center" | 0
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| | if ''b''<sup>2</sup> > ''c''<sup>2</sup> + ''ca'' + ''a''<sup>2</sup> or ''c''<sup>2</sup> > ''a''<sup>2</sup> + ''ab'' + ''b''<sup>2</sup>
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| | (equivalently B > 2π/3 or C > 2π/3)
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| |-
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| | || csc(A + π/3) || otherwise
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| | (equivalently no vertex angle exceeds 2π/3).
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| |}
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| Then ''f'' is bisymmetric and homogeneous so it is a triangle center function. Moreover the corresponding triangle center coincides with the obtuse angled vertex whenever any vertex angle exceeds 2π/3, and with the 1st isogonic center otherwise. Therefore this triangle center is none other than the [[Fermat point]].
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| ==Non-examples==
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| ===Brocard points===
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| The trilinear coordinates of the first Brocard point are ''c''/''b'' : ''a''/''c'' : ''b''/''a''. These coordinates satisfy the properties of homogeneity and cyclicity but not bisymmetry. So the first Brocard point is not (in general) a triangle center. The second Brocard point has trilinear coordinates ''b''/''c'' : ''c''/''a'' : ''a''/''b'' and similar remarks apply.
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| The first and second Brocard points are one of many bicentric pairs of points,<ref>[http://faculty.evansville.edu/ck6/encyclopedia/BicentricPairs.html Bicentric Pairs of Points], Encyclopedia of Triangle Centers, accessed 2012-05-02</ref> pairs of points defined from a triangle with the property that the pair (but not each individual point) is preserved under similarities of the triangle. Several binary operations, such as midpoint and trilinear product, when applied to the two Brocard points, as well as other bicentric pairs, produce triangle centers.
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| ==Some well-known triangle centers==
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| ===Classical triangle centers===
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| <center>
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| {| class="wikitable" border="1"
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| |-
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| ! Position in<br />Encyclopedia of<br /> Triangle Centers
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| ! width="130" | Name
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| ! Notation
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| ! width="350" | Trilinear coordinates
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| |-
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| | align="center" | ''X''<sub>1</sub>
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| | [[Incenter]]
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| | align="center" | ''I''
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| | 1 : 1 : 1
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| |-
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| | align="center" | ''X''<sub>2</sub>
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| | [[Centroid#Of_triangle_and_tetrahedron|Centroid]]
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| | align="center" | ''G''
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| | ''bc'' : ''ca'' : ''ab''
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| |-
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| | align="center" | ''X''<sub>3</sub>
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| | [[Circumcenter]]
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| | align="center" | ''O''
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| | cos ''A'' : cos ''B'' : cos ''C''
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| |-
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| | align="center" | ''X''<sub>4</sub>
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| | [[Orthocenter]]
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| | align="center" | ''H''
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| | sec ''A'' : sec ''B'' : sec ''C''
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| |-
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| | align="center" | ''X''<sub>5</sub>
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| | [[Nine-point center]]
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| | align="center" | ''N''
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| | cos(''B'' − ''C'') : cos(''C'' − ''A'') : cos(''A'' − ''B'')
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| |-
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| | align="center" | ''X''<sub>6</sub>
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| | [[Symmedian point]]
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| | align="center" | ''K''
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| | ''a'' : ''b'' : ''c''
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| |-
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| | align="center" | ''X''<sub>7</sub>
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| | [[Gergonne point#Gergonne triangle and point|Gergonne point]]
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| | align="center" | ''G''<sub>''e''</sub>
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| | ''bc''/(''b'' + ''c'' − ''a'') : ''ca''/(''c'' + ''a'' − ''b'') : ''ab''/(''a'' + ''b'' − ''c'')
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| |-
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| | align="center" | ''X''<sub>8</sub>
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| | [[Nagel point]]
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| | align="center" | ''N''<sub>''a''</sub>
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| | (''b'' + ''c'' − ''a'')/''a'' : (''c'' + ''a'' − ''b'')/''b'': (''a'' + ''b'' − ''c'')/''c''
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| |-
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| | align="center" | ''X''<sub>9</sub>
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| | [[Mittenpunkt]]
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| | align="center" | ''M''
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| | ''b'' + ''c'' − ''a'' : ''c'' + ''a'' − ''b'' : ''a'' + ''b'' − ''c''
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| |-
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| | align="center" | ''X''<sub>10</sub>
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| | [[Spieker center]]
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| | align="center" | ''S''<sub>''p''</sub>
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| | ''bc''(''b'' + ''c'') : ''ca''(''c'' + ''a'') : ''ab''(''a'' + ''b'')
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| |-
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| | align="center" | ''X''<sub>11</sub>
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| | [[Feuerbach point]]
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| | align="center" | ''F''
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| | 1 − cos(''B'' − ''C'') : 1 − cos(''C'' − ''A'') : 1 − cos(''A'' − ''B'')
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| |-
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| | align="center" | ''X''<sub>13</sub>
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| | [[Fermat point]]
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| | align="center" | ''X''
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| | csc(''A'' + π/3) : csc(''B'' + π/3) : csc(''C'' + π/3) *
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| |-
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| | align="center" | ''X''<sub>15</sub><br /> ''X''<sub>16</sub>
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| | [[Isodynamic point]]s
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| | align="center" | ''S''<br />''S''′
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| | sin(''A'' + π/3) : sin(''B'' + π/3) : sin(''C'' + π/3) <br /> sin(''A'' − π/3) : sin(''B'' − π/3) : sin(''C'' − π/3)
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| | align="center" | ''X''<sub>17</sub><br /> ''X''<sub>18</sub>
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| | [[Napoleon points]]
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| | align="center" | ''N''<br />''N''′
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| | sec(''A'' − π/3) : sec(''B'' − π/3) : sec(''C'' − π/3) <br /> sec(''A'' + π/3) : sec(''B'' + π/3) : sec(''C'' + π/3)
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| |-
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| | align="center" | ''X''<sub>99 </sub>
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| | [[Steiner point (triangle)|Steiner point]]
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| | align="center" | ''S''
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| | ''bc''/(''b''<sup>2</sup> − ''c''<sup>2</sup>) : ''ca''/(''c''<sup>2</sup> − ''a''<sup>2</sup>) : ''ab''/(''a''<sup>2</sup> − ''b''<sup>2</sup>)
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| |}
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| (*) : actually the 1st isogonic center, but also the Fermat point whenever ''A'',''B'',''C'' ≤ 2π/3
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| </center>
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| ===Recent triangle centers===
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| In the following table of recent triangle centers, no specific notations are mentioned for the various points.
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| Also for each center only the first trilinear coordinate f(a,b,c) is specified. The other coordinates can be easily derived using the cyclicity property of trilinear coordinates.
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| <center>
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| {| class="wikitable" border="1"
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| ! Position in<br />Encyclopedia of<br /> Triangle Centers
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| ! width="230" | Name
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| ! width="260" | Center function<br />f(a,b,c)
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| |-
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| | align="center" | ''X''<sub>21</sub>
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| | [[Schiffler point]]
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| | 1/(cos ''B'' + cos ''C'')
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| |-
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| | align="center" | ''X''<sub>22 </sub>
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| | [[Exeter point]]
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| | ''a''(''b''<sup>4</sup> + ''c''<sup>4</sup> − ''a''<sup>4</sup>)
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| |-
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| | align="center" | ''X''<sub>111 </sub>
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| | [[Parry point (triangle)|Parry point]]
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| | ''a''/(2''a''<sup>2</sup> − ''b''<sup>2</sup> − ''c''<sup>2</sup>)
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| | align="center" | ''X''<sub>173 </sub>
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| | [[Congruent isoscelizers point]]
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| | tan(''A''/2) + sec(''A''/2)
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| |-
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| | align="center" | ''X''<sub>174 </sub>
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| | [[Yff center of congruence]]
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| | sec(''A''/2)
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| |-
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| | align="center" | ''X''<sub>175 </sub>
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| | [[Isoperimetric point]]
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| | − 1 + sec(''A''/2) cos(''B''/2) cos(''C''/2)
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| |-
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| | align="center" | ''X''<sub>179 </sub>
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| | [[Ajima–Malfatti points|First Ajima-Malfatti point]]
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| | sec<sup>4</sup>(''A''/4)
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| |-
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| | align="center" | ''X''<sub>181 </sub>
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| | [[Apollonius point]]
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| | ''a''(''b'' + ''c'')<sup>2</sup>/(''b'' + ''c'' − ''a'')
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| |-
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| | align="center" | ''X''<sub>192 </sub>
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| | [[Equal parallelians point]]
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| | ''bc''(''ca'' + ''ab'' − ''bc'')
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| |-
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| | align="center" | ''X''<sub>356 </sub>
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| | [[Morley centers|Morley center]]
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| | cos(''A''/3) + 2 cos(''B''/3) cos(''C''/3)
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| |-
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| | align="center" | ''X''<sub>360 </sub>
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| | [[Hofstadter points|Hofstadter point]]
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| | ''A''/''a''
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| |}
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| </center>
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| ==General classes of triangle centers==
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| ===Kimberling center===
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| In honor of Clark Kimberling who created the online encyclopedia of more than 5000 triangle centers, the triangle centers listed in the encyclopedia are collectively called ''Kimberling centers''.<ref>{{cite web|url=http://mathworld.wolfram.com/KimberlingCenter.html|title=Kimberling Center|last=Weisstein|first=Eric W|work=MathWorld–A Wolfram Web Resource. |accessdate=25 May 2009}}</ref>
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| ===Polynomial triangle center===
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| A triangle center P is called a ''polynomial triangle center'' if the trilinear coordinates of P can be expressed as polynomials in ''a'', ''b'' and ''c''.
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| ===Regular triangle center===
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| A triangle center P is called a ''regular triangle point'' if the trilinear coordinates of P can be expressed as polynomials in Δ, ''a'', ''b'' and ''c'', where Δ is the area of the triangle.
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| ===Major triangle center===
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| A triangle center P is said to be a ''major triangle center'' if the trilinear coordinates of P can be expressed in the form f(A) : f(B) : f(C) where f(A) is a function of A alone.<ref>{{cite web|url=http://mathworld.wolfram.com/MajorTriangleCenter.html|title=Major Triangle Center|last=Weisstein|first=Eric W|work=MathWorld–A Wolfram Web Resource|accessdate=25 May 2009}}</ref>
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| ===Transcendental triangle center===
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| A triangle center P is called a ''transcendental triangle center'' if P has no trilinear representation using only algebraic functions of a, b and c.
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| ==Miscellaneous==
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| ===Isosceles and equilateral triangles===
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| Let ''f'' be a triangle center function. If two sides of a triangle are equal (say ''a'' = ''b'') then
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| :''f''(''a'', ''b'', ''c'') = ''f''(''b'', ''a'', ''c'')     since ''a'' = ''b''
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| ::: = ''f''(''b'', ''c'', ''a'')     by bisymmetry
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| so two components of the associated triangle center are always equal. Therefore all triangle centers of an isosceles triangle must lie on its line of symmetry. For an equilateral triangle all three components are equal so all centers coincide with the centroid. So, like a circle, an equilateral triangle has a unique center.
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| ===Excenters===
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| {|
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| |-
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| | rowspan=2 | Let ''f''(''a'',''b'',''c'') =
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| | rowspan=2 | <math>\begin{cases} \\ \end{cases}</math> || −1
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| | if ''a'' ≥ ''b'' and ''a'' ≥ ''c''
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| |-
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| | align="center" | 1
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| | otherwise
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| |}
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| This is readily seen to be a triangle center function and (provided the triangle is scalene) the corresponding triangle center is the excenter opposite to the largest vertex angle. The other two excenters can be picked out by similar functions. However as indicated above only one of the excenters of an isosceles triangle and none of the excenters of an equilateral triangle can ever be a triangle center.
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| ===Biantisymmetric functions===
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| A function ''f'' is '''biantisymmetric''' if ''f''(''a'',''b'',''c'') = −''f''(''a'',''c'',''b'') for all ''a'',''b'',''c''. If such a function is also non-zero and homogeneous it is easily seen that the mapping (a,b,c) → ''f''(''a'',''b'',''c'')<sup>2</sup> ''f''(''b'',''c'',''a'') ''f''(''c'',''a'',''b'') is a triangle center function. The corresponding triangle center is ''f''(''a'',''b'',''c'') : ''f''(''b'',''c'',''a'') : ''f''(''c'',''a'',''b''). On account of this the definition of triangle center function is sometimes taken to include non-zero homogeneous biantisymmetric functions.
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| ===New centers from old===
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| Any triangle center function ''f'' can be '''normalized''' by multiplying it by a symmetric function of ''a'',''b'',''c'' so that ''n'' = 0. A normalized triangle center function has the same triangle center as the original, and also the stronger property that ''f''(''ta'',''tb'',''tc'') = ''f''(''a'',''b'',''c'') for all ''t'' > 0 and all (''a'',''b'',''c''). Together with the zero function, normalized triangle center functions form an [[algebra over a field|algebra]] under addition, subtraction, and multiplication. This gives an easy way to create new triangle centers. However distinct normalized triangle center functions will often define the same triangle center, for example ''f'' and (''abc'')<sup>−1</sup>(''a''+''b''+''c'')<sup>3</sup>''f'' .
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| ===Uninteresting centers===
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| Assume ''a'',''b'',''c'' are real variables and let α,β,γ be any three real constants.
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| {|
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| |-
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| | rowspan=3 | <math>\begin{cases} \\ \\ \end{cases}</math> || α
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| | if ''a'' < ''b'' and ''a'' < ''c''
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| | (equivalently the first variable is the smallest)
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| |-
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| | Let ''f''(''a'',''b'',''c'') =
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| | align="center" | γ
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| | if ''a'' > ''b'' and ''a'' > ''c''
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| | (equivalently the first variable is the largest)
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| |-
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| | align="center" | β || otherwise
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| | (equivalently the first variable is in the middle)
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| |}
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| Then ''f'' is a triangle center function and α : β : γ is the corresponding triangle center whenever the sides of the reference triangle are labelled so that ''a'' < ''b'' < ''c''. Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest. The [[Encyclopedia of Triangle Centers]] is an ever-expanding list of interesting ones.
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| ===Barycentric coordinates===
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| If ''f'' is a triangle center function then so is ''af'' and the corresponding triangle center is ''af''(''a'',''b'',''c'') : ''bf''(''b'',''c'',''a'') : ''cf''(''c'',''a'',''b''). Since these are precisely the [[Barycentric coordinate system|barycentric coordinates]] of the triangle center corresponding to ''f'' it follows that triangle centers could equally well have been defined in terms of barycentrics instead of trilinears. In practice it isn't difficult to switch from one coordinate system to the other.
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| ===Binary systems===
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| There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed by ''X''<sub>3</sub> and the incenter of the tangential triangle. Consider the triangle center function given by :
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| {|
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| |-
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| | rowspan=3 | <math>\begin{cases} \\ \\ \end{cases}</math> || align="center" | cos(''A'')
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| | if the triangle is acute.
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| |-
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| | ''f''(''a'',''b'',''c'') =
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| | cos(''A'') + sec(''B'')sec(''C'')
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| | if the vertex angle at ''A'' is obtuse.
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| |-
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| | || align="center" | cos(''A'') − sec(''A'')
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| | if either of the angles at ''B'' or ''C'' is obtuse.
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| |}
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| For the corresponding triangle center there are four distinct possibilities:
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| :* cos(''A'') : cos(''B'') : cos(''C'') if the reference triangle is acute (this is also the circumcenter).
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| :* cos(''A'') + sec(''B'')sec(''C'') : cos(''B'') − sec(''B'') : cos(''C'') − sec(''C'') if the angle at ''A'' is obtuse.
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| :* cos(''A'') − sec(''A'') : cos(''B'') + sec(''C'')sec(''A'') : cos(''C'') − sec(''C'') if the angle at ''B'' is obtuse.
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| :* cos(''A'') − sec(''A'') : cos(''B'') − sec(''B'') : cos(''C'') + sec(''A'')sec(''B'') if the angle at ''C'' is obtuse.
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| Routine calculation shows that in every case these trilinears represent the incenter of the tangential triangle. So this point is a triangle center that is a close companion of the circumcenter.
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| ===Bisymmetry and invariance===
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| Reflecting a triangle reverses the order of its sides. In the image the coordinates refer to the (''c'',''b'',''a'') triangle and (using "|" as the separator) the reflection of an arbitrary point α : β : γ is γ | β | α. If ''f'' is a triangle center function the reflection of its triangle center is ''f''(''c'',''a'',''b'') | ''f''(''b'',''c'',''a'') | ''f''(''a'',''b'',''c'') which, by bisymmetry, is the same as ''f''(''c'',''b'',''a'') | ''f''(''b'',''a'',''c'') | ''f''(''a'',''c'',''b''). As this is also the triangle center corresponding to ''f'' relative to the (''c'',''b'',''a'') triangle, bisymmetry ensures that all triangle centers are invariant under reflection. Since rotations and translations may be regarded as double reflections they too must preserve triangle centers. These invariance properties provide justification for the definition.
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| ===Alternative terminology===
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| Some other names for dilation are [[scaling (geometry)|uniform scaling]], [[scaling (geometry)|isotropic scaling]], [[homothetic transformation|homothety]], and [[homothetic transformation|homothecy]].
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| ==Hyperbolic triangle centers==
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| The study of triangle centers traditionally is concerned with [[Euclidean geometry]], but triangle centers can also be studied in [[hyperbolic geometry]]. Using [[gyrotrigonometry]], expressions for trigonometric barycentric coordinates can be calculated that have the same form for both euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must ''not'' encapsulate the specification of the anglesum being 180 degrees.<ref>[http://ajmaa.org/searchroot/files/pdf/v6n1/v6i1p18.pdf Hyperbolic Barycentric Coordinates],
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| Abraham A. Ungar, The Australian Journal of Mathematical Analysis and Applications, AJMAA, Volume 6, Issue 1, Article 18, pp. 1-35, 2009</ref><ref>[http://www.springer.com/astronomy/cosmology/book/978-90-481-8636-5 Hyperbolic Triangle Centers: The Special Relativistic Approach], Abraham Ungar, Springer, 2010</ref><ref name="barycalc">[http://www.worldscibooks.com/mathematics/7740.html Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction], Abraham Ungar, World Scientific, 2010</ref>
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| ==Tetrahedron centers and n-simplex centers==
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| A generalization of triangle centers to higher dimensions is centers of [[tetrahedron]]s or higher-dimensional [[simplex|simplices]].<ref name="barycalc"/>
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| ==See also==
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| *[[Central line (geometry)|Central line]]
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| *[[Encyclopedia of Triangle Centers]]
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| ==Notes==
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| <references />
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| ==External links==
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| *For detailed descriptions and nice diagrams of certain specific triangle centers: MathWorld–A Wolfram Web Resource: {{cite web|url=http://mathworld.wolfram.com/topics/TriangleCenters.html|title=Triangle Centers|last=Weisstein|first=Eric W|accessdate=23 May 2009}}
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| *For a discussion on the distribution of triangle centers: [http://www.paideiaschool.org/Teacherpages/Steve_Sigur/resources/all%20points%20web/all-points.html The Triangle is a Busy Place – The Distribution of Triangle Centers Project] (Accessed on 25 May 2009)
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| *URL of Clark Kimberling's Encyclopedia of Triangle centers: [http://faculty.evansville.edu/ck6/encyclopedia/ ETC]
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| *[http://www.dekovsoft.com/e.htm Computer-Generated Encyclopedia of Euclidean Geometry] The first part of the encyclopedia contains more than 3000 computer-generated statements of theorems in Triangle Geometry.
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| *A list of links to geometry pages in the internet: [http://web.archive.org/web/20091027004020/http://de.geocities.com/darij_grinberg/Links.html Links on Geometry]
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| [[Category:Triangles]]
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| [[Category:Triangle centers|*]]
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| [[Category:Triangle geometry]]
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