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Triangle center

From Wikipedia, the free encyclopedia
Point in a triangle that can be seen as its middle under some criteria
This article is about a geometry concept. For the office and retail complex, seeTriangle Center.

Five important triangle centers.
  Reference triangle△ABC
  Angle bisectors andincircle (intersect/centered atincenterI)
  Medians (intersect atcentroidG)
  Perpendicular bisectors andcircumcircle (intersect/centered at circumcenterO)
  Altitudes (intersect atorthocenterH)
  Nine-point circle (centered atnine-point centerN which, along withH, G, O, lies on theEuler linee)

Ingeometry, atriangle center ortriangle centre is apoint in thetriangle'splane that is in some sense in the middle of the triangle. For example, thecentroid,circumcenter,incenter andorthocenter were familiar to theancient Greeks, and can be obtained by simpleconstructions.

Each of these classical centers has the property that it isinvariant (more preciselyequivariant) undersimilarity transformations. In other words, for any triangle and any similarity transformation (such as arotation,reflection,dilation, ortranslation), the center of the transformed triangle is the same point as the transformed center of the original triangle.This invariance is the defining property of a triangle center. It rules out other well-known points such as theBrocard points which are not invariant under reflection and so fail to qualify as triangle centers.

For anequilateral triangle, all triangle centers coincide at its centroid. However, the triangle centers generally take different positions from each other on all other triangles. The definitions and properties of tens of thousands of triangle centers have been collected in theEncyclopedia of Triangle Centers.

History

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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 theFermat point,nine-point center,Lemoine point,Gergonne point, andFeuerbach 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.[1][2]Clark Kimberling'sEncyclopedia of Triangle Centers contains an annotated list of over 50,000 triangle centers.[3] Every entry in theEncyclopedia of Triangle Centers is denoted byX(n){\displaystyle X(n)} orXn{\displaystyle X_{n}} wheren{\displaystyle n} is the positional index of the entry. For example, thecentroid of a triangle is the second entry and is denoted byX(2){\displaystyle X(2)} orX2{\displaystyle X_{2}}.

Formal definition

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Areal-valued functionf of three real variablesa, b, c may have the following properties:

If a non-zerof has both these properties it is called a triangle center function. Iff is a triangle center function anda, b, c are the side-lengths of a reference triangle then the point whosetrilinear coordinates aref(a,b,c):f(b,c,a):f(c,a,b){\displaystyle f(a,b,c):f(b,c,a):f(c,a,b)} is called a triangle center.

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 bycyclic permutation ofa, b, c. This process is known ascyclicity.[4][5]

Every triangle center function corresponds to a unique triangle center. This correspondence is notbijective. Different functions may define the same triangle center. For example, the functionsf1(a,b,c)=1a{\displaystyle f_{1}(a,b,c)={\tfrac {1}{a}}} andf2(a,b,c)=bc{\displaystyle f_{2}(a,b,c)=bc} both correspond to the centroid.Two triangle center functions define the same triangle center if and only if their ratio is a function symmetric ina, b, c.

Even if a triangle center function is well-defined everywhere the same cannot always be said for its associated triangle center. For example, letf(a,b,c){\displaystyle f(a,b,c)} be 0 if⁠ab{\displaystyle {\tfrac {a}{b}}}⁠ and⁠ac{\displaystyle {\tfrac {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.

Default domain

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In some cases these functions are not defined on the whole of⁠R3.{\displaystyle \mathbb {R} ^{3}.}⁠ For example, the trilinears ofX365 which is the 365th entry in theEncyclopedia of Triangle Centers, area1/2:b1/2:c1/2{\displaystyle a^{1/2}:b^{1/2}:c^{1/2}} soa, b, c cannot be negative. Furthermore, in order to represent the sides of a triangle they must satisfy thetriangle inequality. So, in practice, every function'sdomain is restricted to the region of⁠R3{\displaystyle \mathbb {R} ^{3}}⁠ wherea≤b+c,b≤c+a,c≤a+b.{\displaystyle a\leq b+c,\quad b\leq c+a,\quad c\leq a+b.}This regionT is the domain of all triangles, and it is the default domain for all triangle-based functions.

Other useful domains

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There are various instances where it may be desirable to restrict the analysis to a smaller domain thanT. For example:

a2>b2+bc+c2;b2>c2+ca+a2;c2>a2+ab+b2.{\displaystyle a^{2}>b^{2}+bc+c^{2};\quad b^{2}>c^{2}+ca+a^{2};\quad c^{2}>a^{2}+ab+b^{2}.}

  • A domain of much practical value since it is dense inT yet excludes all trivial triangles (i.e. points) and degenerate triangles (i.e. lines) is the set of allscalene triangles. It is obtained by removing the planesb =c,c =a,a =b fromT.

Domain symmetry

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Not every subsetD ⊆T is a viable domain. In order to support the bisymmetry testD must be symmetric about the planesb =c,c =a,a =b. To support cyclicity it must also be invariant under 2π/3 rotations about the linea =b =c. The simplest domain of all is the line(t,t,t) which corresponds to the set of allequilateral triangles.

Examples

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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

a(b2+c2−a2):b(c2+a2−b2):c(a2+b2−c2).{\displaystyle a(b^{2}+c^{2}-a^{2}):b(c^{2}+a^{2}-b^{2}):c(a^{2}+b^{2}-c^{2}).}

Letf(a,b,c)=a(b2+c2−a2){\displaystyle f\left(a,b,c\right)=a\left(b^{2}+c^{2}-a^{2}\right)} It can be shown thatf is homogeneous:f(ta,tb,tc)=ta[(tb)2+(tc)2−(ta)2]=t3[a(b2+c2−a2)]=t3f(a,b,c){\displaystyle {\begin{aligned}f(ta,tb,tc)&=ta{\Bigl [}(tb)^{2}+(tc)^{2}-(ta)^{2}{\Bigr ]}\\[2pt]&=t^{3}{\Bigl [}a(b^{2}+c^{2}-a^{2}){\Bigr ]}\\[2pt]&=t^{3}f(a,b,c)\end{aligned}}}as well as bisymmetric:f(a,c,b)=a(c2+b2−a2)=a(b2+c2−a2)=f(a,b,c){\displaystyle {\begin{aligned}f(a,c,b)&=a(c^{2}+b^{2}-a^{2})\\[2pt]&=a(b^{2}+c^{2}-a^{2})\\[2pt]&=f(a,b,c)\end{aligned}}}sof 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.

1st isogonic center

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Let△A'BC be the equilateral triangle having baseBC and vertexA' on the negative side ofBC and let△AB'C and△ABC' be similarly constructed equilateral triangles based on the other two sides of triangle△ABC. Then the linesAA', BB', CC' are concurrent and the point of concurrence is the 1st isogonal center. Its trilinear coordinates are

csc⁡(A+π3):csc⁡(B+π3):csc⁡(C+π3).{\displaystyle \csc \left(A+{\frac {\pi }{3}}\right):\csc \left(B+{\frac {\pi }{3}}\right):\csc \left(C+{\frac {\pi }{3}}\right).}

Expressing these coordinates in terms ofa, b, 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.

Fermat point

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Let

f(a,b,c)={1if a2>b2+bc+c2⟺if A>2π/30if b2>c2+ca+a2 or c2>a2+ab+b2⟺if B>2π/3 or C>2π/3csc⁡(A+π3)otherwise ⟺A,B,C≤2π/3{\displaystyle f(a,b,c)={\begin{cases}1&\quad {\text{if }}a^{2}>b^{2}+bc+c^{2}&\iff {\text{if }}A>2\pi /3\\[8pt]0&\quad \!\!\displaystyle {{{\text{if }}b^{2}>c^{2}+ca+a^{2}} \atop {{\text{ or }}c^{2}>a^{2}+ab+b^{2}}}&\iff \!\!\displaystyle {{{\text{if }}B>2\pi /3} \atop {{\text{ or }}C>2\pi /3}}\\[8pt]\csc(A+{\frac {\pi }{3}})&\quad {\text{otherwise }}&\iff A,B,C\leq 2\pi /3\end{cases}}}

Thenf 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 theFermat point.

Non-examples

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Brocard points

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Main article:Brocard points

The trilinear coordinates of the first Brocard point are:cb : ac : ba{\displaystyle {\frac {c}{b}}\ :\ {\frac {a}{c}}\ :\ {\frac {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:bc : ca : ab{\displaystyle {\frac {b}{c}}\ :\ {\frac {c}{a}}\ :\ {\frac {a}{b}}}and similar remarks apply.

The first and second Brocard points are one of many bicentric pairs of points,[6] 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.

Some well-known triangle centers

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Classical triangle centers

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ETC reference;
Name; Symbol		Trilinear coordinatesDescription
X1IncenterI1:1:1{\displaystyle 1:1:1}Intersection of theangle bisectors. Center of the triangle'sinscribed circle.
X2CentroidGbc:ca:ab{\displaystyle bc:ca:ab}Intersection of themedians.Center of mass of a uniform triangularlamina.
X3CircumcenterOcos⁡A:cos⁡B:cos⁡C{\displaystyle \cos A:\cos B:\cos C}Intersection of theperpendicular bisectors of the sides. Center of the triangle'scircumscribed circle.
X4OrthocenterHsec⁡A:sec⁡B:sec⁡C{\displaystyle \sec A:\sec B:\sec C}Intersection of thealtitudes.
X5Nine-point centerNcos⁡(B−C):cos⁡(C−A):cos⁡(A−B){\displaystyle \cos(B-C):\cos(C-A):\cos(A-B)}Center of the circle passing through the midpoint of each side, the foot of each altitude, and the midpoint between the orthocenter and each vertex.
X6Symmedian pointKa:b:c{\displaystyle a:b:c}Intersection of the symmedians – the reflection of each median about the corresponding angle bisector.
X7Gergonne pointGebcb+c−a:cac+a−b:aba+b−c{\displaystyle {\frac {bc}{b+c-a}}:{\frac {ca}{c+a-b}}:{\frac {ab}{a+b-c}}}Intersection of the lines connecting each vertex to the point where the incircle touches the opposite side.
X8Nagel pointNab+c−aa:c+a−bb:a+b−cc{\displaystyle {\frac {b+c-a}{a}}:{\frac {c+a-b}{b}}:{\frac {a+b-c}{c}}}Intersection of the lines connecting each vertex to the point where an excircle touches the opposite side.
X9MittenpunktM(b+c−a):(c+a−b):(a+b−c){\displaystyle (b+c-a):(c+a-b):(a+b-c)}Symmedian point of theexcentral triangle (and various equivalent definitions).
X10Spieker centerSpbc(b+c):ca(c+a):ab(a+b){\displaystyle bc(b+c):ca(c+a):ab(a+b)}Incenter of the medial triangle. Center of mass of a uniform triangular wireframe.
X11Feuerbach pointF1−cos⁡(B−C):1−cos⁡(C−A):1−cos⁡(A−B){\displaystyle 1-\cos(B-C):1-\cos(C-A):1-\cos(A-B)}Point at which the nine-point circle is tangent to the incircle.
X13Fermat pointXcsc⁡(A+π3):csc⁡(B+π3):csc⁡(C+π3).{\displaystyle \csc(A+{\tfrac {\pi }{3}}):\csc(B+{\tfrac {\pi }{3}}):\csc(C+{\tfrac {\pi }{3}}).}[a]Point that is the smallest possible sum of distances from the vertices.
X15
X16		Isodynamic pointsS
S′		sin⁡(A+π3):sin⁡(B+π3):sin⁡(C+π3)sin⁡(A−π3):sin⁡(B−π3):sin⁡(C−π3){\displaystyle {\begin{aligned}\sin(A+{\tfrac {\pi }{3}}):\sin(B+{\tfrac {\pi }{3}}):\sin(C+{\tfrac {\pi }{3}})\\\sin(A-{\tfrac {\pi }{3}}):\sin(B-{\tfrac {\pi }{3}}):\sin(C-{\tfrac {\pi }{3}})\end{aligned}}}Centers ofinversion that transform the triangle into an equilateral triangle.
X17
X18		Napoleon pointsN
N′		sec⁡(A−π3):sec⁡(B−π3):sec⁡(C−π3)sec⁡(A+π3):sec⁡(B+π3):sec⁡(C+π3){\displaystyle {\begin{aligned}\sec(A-{\tfrac {\pi }{3}}):\sec(B-{\tfrac {\pi }{3}}):\sec(C-{\tfrac {\pi }{3}})\\\sec(A+{\tfrac {\pi }{3}}):\sec(B+{\tfrac {\pi }{3}}):\sec(C+{\tfrac {\pi }{3}})\end{aligned}}}Intersection of the lines connecting each vertex to the center of an equilateral triangle pointed outwards (first Napoleon point) or inwards (second Napoleon point), mounted on the opposite side.
X99Steiner pointSbcb2−c2:cac2−a2:aba2−b2{\displaystyle {\frac {bc}{b^{2}-c^{2}}}:{\frac {ca}{c^{2}-a^{2}}}:{\frac {ab}{a^{2}-b^{2}}}}Various equivalent definitions.

Recent triangle centers

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In the following table of more recent triangle centers, no specific notations are mentioned for the various points.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.

ETC reference; NameCenter function
f(a,b,c){\displaystyle f(a,b,c)}		Year described
X21Schiffler point1cos⁡B+cos⁡C{\displaystyle {\frac {1}{\cos B+\cos C}}}1985
X22Exeter pointa(b4+c4−a4){\displaystyle a(b^{4}+c^{4}-a^{4})}1986
X111Parry pointa2a2−b2−c2{\displaystyle {\frac {a}{2a^{2}-b^{2}-c^{2}}}}early 1990s
X173Congruent isoscelizers pointtan⁡A2+sec⁡A2{\displaystyle \tan {\tfrac {A}{2}}+\sec {\tfrac {A}{2}}}1989
X174Yff center of congruencesec⁡A2{\displaystyle \sec {\tfrac {A}{2}}}1987
X175Isoperimetric pointsec⁡A2cos⁡B2cos⁡C2−1{\displaystyle \sec {\tfrac {A}{2}}\cos {\tfrac {B}{2}}\cos {\tfrac {C}{2}}-1}1985
X179First Ajima-Malfatti pointsec4⁡A4{\displaystyle \sec ^{4}{\tfrac {A}{4}}}
X181Apollonius pointa(b+c)2b+c−a{\displaystyle {\frac {a(b+c)^{2}}{b+c-a}}}1987
X192Equal parallelians pointbc(ca+ab−bc){\displaystyle bc(ca+ab-bc)}1961
X356Morley centercos⁡A3+2cos⁡B3cos⁡C3{\displaystyle \cos {\tfrac {A}{3}}+2\cos {\tfrac {B}{3}}\cos {\tfrac {C}{3}}}1978[7]
X360Hofstadter zero pointAa{\displaystyle {\frac {A}{a}}}1992

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 32,000 triangle centers, the triangle centers listed in the encyclopedia are collectively calledKimberling centers.[8]

Polynomial triangle center

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A triangle centerP is called apolynomial triangle center if the trilinear coordinates ofP can be expressed as polynomials ina, b, c.

Regular triangle center

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A triangle centerP is called aregular triangle point if the trilinear coordinates ofP can be expressed as polynomials in△,a,b,c, where△ is the area of the triangle.

Major triangle center

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A triangle centerP is said to be amajor triangle center if the trilinear coordinates of P can be expressed in the formf(A):f(B):f(C){\displaystyle f(A):f(B):f(C)} where⁠f(X){\displaystyle f(X)}⁠ is a function of the angleX alone and does not depend on the other angles or on the side lengths.[9]

Transcendental triangle center

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A triangle centerP is called atranscendental triangle center ifP has no trilinear representation using only algebraic functions ofa, b, c.

Miscellaneous

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Isosceles and equilateral triangles

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Letf be a triangle center function. If two sides of a triangle are equal (saya =b) thenf(a,b,c)=f(b,a,c)(since a=b)=f(b,c,a)(by bisymmetry){\displaystyle {\begin{aligned}f(a,b,c)&=f(b,a,c)&({\text{since }}a=b)\\&=f(b,c,a)&{\text{(by bisymmetry)}}\end{aligned}}}so two components of the associated triangle center are always equal. Therefore, all triangle centers of an isosceles triangle must lie on itsline 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.

Excenters

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Letf(a,b,c)={−1if a≥b and a≥c,1otherwise.{\displaystyle f(a,b,c)={\begin{cases}-1&\quad {\text{if }}a\geq b{\text{ and }}a\geq c,\\\;\;\;1&\quad {\text{otherwise}}.\end{cases}}}

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.

Biantisymmetric functions

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A functionf isbiantisymmetric iff(a,b,c)=−f(a,c,b)for alla,b,c.{\displaystyle f(a,b,c)=-f(a,c,b)\quad {\text{for all}}\quad 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)2f(b,c,a)f(c,a,b){\displaystyle (a,b,c)\to f(a,b,c)^{2}\,f(b,c,a)\,f(c,a,b)}is a triangle center function. The corresponding triangle center isf(a,b,c):f(b,c,a):f(c,a,b).{\displaystyle 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.

New centers from old

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Any triangle center functionf can benormalized by multiplying it by asymmetric function ofa, b, c so thatn = 0. A normalized triangle center function has the same triangle center as the original, and also the stronger property thatf(ta,tb,tc)=f(a,b,c)for allt>0, (a,b,c).{\displaystyle f(ta,tb,tc)=f(a,b,c)\quad {\text{for all}}\quad t>0,\ (a,b,c).}Together with the zero function, normalized triangle center functions form analgebra 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 examplef and(abc)−1(a+b+c)3f.{\displaystyle (abc)^{-1}(a+b+c)^{3}f.}

Uninteresting centers

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Assumea, b, c are real variables and letα, β, γ be any three real constants. Let

f(a,b,c)={αif a<b and a<c(a is least),γif a>b and a>c(a is greatest),βotherwise(a is in the middle).{\displaystyle f(a,b,c)={\begin{cases}\alpha &\quad {\text{if }}a<b{\text{ and }}a<c&(a{\text{ is least}}),\\[2pt]\gamma &\quad {\text{if }}a>b{\text{ and }}a>c&(a{\text{ is greatest}}),\\[2pt]\beta &\quad {\text{otherwise}}&(a{\text{ is in the middle}}).\end{cases}}}

Thenf is a triangle center function andα :β :γ is the corresponding triangle center whenever the sides of the reference triangle are labelled so thata <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.

Barycentric coordinates

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Iff is a triangle center function then so isaf and the corresponding triangle center isaf(a,b,c):bf(b,c,a):cf(c,a,b).{\displaystyle a\,f(a,b,c):b\,f(b,c,a):c\,f(c,a,b).} Since these are precisely thebarycentric coordinates of the triangle center corresponding tof 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.

Binary systems

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There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed byX3 and the incenter of thetangential triangle. Consider the triangle center function given by:

f(a,b,c)={cos⁡Aif △ is acute,cos⁡A+sec⁡Bsec⁡Cif ∡A is obtuse,cos⁡A−sec⁡Aif either∡B or ∡C is obtuse.{\displaystyle f(a,b,c)={\begin{cases}\cos A&{\text{if }}\triangle {\text{ is acute}},\\[2pt]\cos A+\sec B\sec C&{\text{if }}\measuredangle A{\text{ is obtuse}},\\[2pt]\cos A-\sec A&{\text{if either}}\measuredangle B{\text{ or }}\measuredangle C{\text{ is obtuse}}.\end{cases}}}

For the corresponding triangle center there are four distinct possibilities:if reference △ is acute:cos⁡A :cos⁡B :cos⁡Cif ∡A is obtuse:cos⁡A+sec⁡Bsec⁡C:cos⁡B−sec⁡B:cos⁡C−sec⁡Cif ∡B is obtuse:cos⁡A−sec⁡A:cos⁡B+sec⁡Csec⁡A:cos⁡C−sec⁡Cif ∡C is obtuse:cos⁡A−sec⁡A:cos⁡B−sec⁡B:cos⁡C+sec⁡Asec⁡B{\displaystyle {\begin{aligned}&{\text{if reference }}\triangle {\text{ is acute:}}\quad \cos A\ :\,\cos B\ :\,\cos C\\[6pt]&{\begin{array}{rcccc}{\text{if }}\measuredangle A{\text{ is obtuse:}}&\cos A+\sec B\sec C&:&\cos B-\sec B&:&\cos C-\sec C\\[4pt]{\text{if }}\measuredangle B{\text{ is obtuse:}}&\cos A-\sec A&:&\cos B+\sec C\sec A&:&\cos C-\sec C\\[4pt]{\text{if }}\measuredangle C{\text{ is obtuse:}}&\cos A-\sec A&:&\cos B-\sec B&:&\cos C+\sec A\sec B\end{array}}\end{aligned}}}Note that the first is also the circumcenter.

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.

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 pointγ:β:α{\displaystyle \gamma :\beta :\alpha } isγ | β | α.{\displaystyle \gamma \ |\ \beta \ |\ \alpha .} Iff is a triangle center function the reflection of its triangle center isf(c,a,b) | f(b,c,a) | f(a,b,c),{\displaystyle f(c,a,b)\ |\ f(b,c,a)\ |\ f(a,b,c),} which, by bisymmetry, is the same asf(c,b,a) | f(b,a,c) | f(a,c,b).{\displaystyle f(c,b,a)\ |\ f(b,a,c)\ |\ f(a,c,b).} As this is also the triangle center corresponding tof 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.

Alternative terminology

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Some other names for dilation areuniform scaling,isotropic scaling,homothety, andhomothecy.

Non-Euclidean and other geometries

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The study of triangle centers traditionally is concerned withEuclidean geometry, but triangle centers can also be studied innon-Euclidean geometry.[10] Triangle centers that have the same form for both Euclidean andhyperbolic geometry can be expressed usinggyrotrigonometry.[11][12][13] In non-Euclidean geometry, the assumption that the interior angles of the triangle sum to 180 degrees must be discarded.

Centers oftetrahedra or higher-dimensionalsimplices can also be defined, by analogy with 2-dimensional triangles.[13]

Some centers can be extended to polygons with more than three sides. Thecentroid, for instance, can be found for any polygon. Some research has been done on the centers of polygons with more than three sides.[14][15]

See also

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Notes

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  1. ^actually the 1st isogonic center, but also the Fermat point wheneverA,B,C ≤ 2π/3
  1. ^Kimberling, Clark."Triangle centers". Retrieved2009-05-23.Unlike squares and circles, triangles have many centers. The ancient Greeks found four: incenter, centroid, circumcenter, and orthocenter. A fifth center, found much later, is the Fermat point. Thereafter, points now called nine-point center, symmedian point, Gergonne point, and Feuerbach point, to name a few, were added to the literature. 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
  2. ^Kimberling, Clark (11 Apr 2018) [1994]. "Central Points and Central Lines in the Plane of a Triangle".Mathematics Magazine.67 (3):163–187.doi:10.2307/2690608.JSTOR 2690608.
  3. ^Kimberling, Clark."This is PART 26: Centers X(50001) – X(52000)".Encyclopedia of Triangle Centers. Retrieved17 June 2022.
  4. ^Weisstein, Eric W."Triangle Center".MathWorld–A Wolfram Web Resource. Retrieved25 May 2009.
  5. ^Weisstein, Eric W."Triangle Center Function".MathWorld–A Wolfram Web Resource. Retrieved1 July 2009.
  6. ^Bicentric Pairs of Points, Encyclopedia of Triangle Centers, accessed 2012-05-02
  7. ^Oakley, Cletus O.; Baker, Justine C. (November 1978)."The Morley Trisector Theorem".The American Mathematical Monthly.85 (9):737–745.doi:10.1080/00029890.1978.11994688.ISSN 0002-9890.
  8. ^Weisstein, Eric W."Kimberling Center".MathWorld–A Wolfram Web Resource. Retrieved25 May 2009.
  9. ^Weisstein, Eric W."Major Triangle Center".MathWorld–A Wolfram Web Resource. Retrieved25 May 2009.
  10. ^Russell, Robert A. (2019-04-18). "Non-Euclidean Triangle Centers".arXiv:1608.08190 [math.MG].
  11. ^Ungar, Abraham A. (2009)."Hyperbolic Barycentric Coordinates"(PDF).The Australian Journal of Mathematical Analysis and Applications.6 (1):1–35., article #18
  12. ^Ungar, Abraham A. (2010).Hyperbolic triangle centers : the special relativistic approach. Dordrecht: Springer.ISBN 978-90-481-8637-2.OCLC 663096629.
  13. ^abUngar, Abraham Albert (August 2010).Barycentric Calculus in Euclidean and Hyperbolic Geometry. WORLD SCIENTIFIC.doi:10.1142/7740.ISBN 978-981-4304-93-1.
  14. ^Al-Sharif, Abdullah; Hajja, Mowaffaq; Krasopoulos, Panagiotis T. (November 2009)."Coincidences of Centers of Plane Quadrilaterals".Results in Mathematics.55 (3–4):231–247.doi:10.1007/s00025-009-0417-6.ISSN 1422-6383.S2CID 122725235.
  15. ^Prieto-Martínez, Luis Felipe; Sánchez-Cauce, Raquel (2021-04-02)."Generalization of Kimberling's Concept of Triangle Center for Other Polygons".Results in Mathematics.76 (2): 81.arXiv:2004.01677.doi:10.1007/s00025-021-01388-4.ISSN 1420-9012.S2CID 214795185.

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