In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a triangle center (or triangle centre) is a point in the plane that is in some sense a
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of a
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
akin to the centers of
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s, that is, a point that is in the middle of the figure by some measure. For example, the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
,
circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
,
incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
and
orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
were familiar to the
ancient Greeks
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cultu ...
, and can be obtained by simple
constructions.
Each of these classical centers has the property that it is
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
(more precisely
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
) under
similarity transformations. In other words, for any triangle and any similarity transformation (such as a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
,
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
,
dilation
Dilation (or dilatation) may refer to:
Physiology or medicine
* Cervical dilation, the widening of the cervix in childbirth, miscarriage etc.
* Coronary dilation, or coronary reflex
* Dilation and curettage, the opening of the cervix and surgic ...
, or
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
), 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 the
Brocard points
In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician.
Definition
In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...
which are not invariant under reflection and so fail to qualify as triangle centers.
For an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
, 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 thousands of triangle centers have been collected in the ''
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the ...
''.
History
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
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...
,
nine-point center
In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle.
It is so called because it is the center of the nine-point circle, a circle t ...
,
Lemoine point
In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle.
Ross Honsberger called its existence "one of the crown jewels of m ...
,
Gergonne point
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
, and
Feuerbach point
In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on th ...
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. ,
Clark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...
's ''
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the ...
'' contains an annotated list of 50,730 triangle centers. Every entry in the ''Encyclopedia of Triangle Centers'' is denoted by
or
where
is the positional index of the entry. For example, the
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
of a triangle is the second entry and is denoted by
or
.
Formal definition
A
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real f ...
''f'' of three real variables ''a'', ''b'', ''c'' may have the following properties:
*Homogeneity: ''f''(''ta'',''tb'',''tc'') = ''t''
''n'' ''f''(''a'',''b'',''c'') for some constant ''n'' and for all ''t'' > 0.
*Bisymmetry in the second and third variables: ''f''(''a'',''b'',''c'') = ''f''(''a'',''c'',''b'').
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
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
are ''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 by
cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
of ''a'', ''b'', ''c''. This process is known as cyclicity.
Every triangle center function corresponds to a unique triangle center. This correspondence is not
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
. Different functions may define the same triangle center. For example, the functions ''f''
1(''a'',''b'',''c'') = 1/''a'' and ''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 in ''a'', ''b'' and ''c''.
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.
Default domain
In some cases these functions are not defined on the whole of
ℝ3. For example, the trilinears of ''X''
365 which is the 365th entry in the
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the ...
, are ''a''
1/2 : ''b''
1/2 : ''c''
1/2 so ''a'', ''b'', ''c'' cannot be negative. Furthermore, in order to represent the sides of a triangle they must satisfy the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
. So, in practice, every function's
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
is restricted to the region of
ℝ3 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.
Other useful domains
There are various instances where it may be desirable to restrict the analysis to a smaller domain than T. For example:
:*The centers ''X''
3, ''X''
4, ''X''
22, ''X''
24, ''X''
40 make specific reference to
acute triangle
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
s,
namely that region of T where ''a''
2 ≤ ''b''
2 + ''c''
2, ''b''
2 ≤ ''c''
2 + ''a''
2, ''c''
2 ≤ ''a''
2 + ''b''
2.
:*When differentiating between the Fermat point and ''X''
13 the domain of triangles with an angle exceeding 2π/3 is important,
in other words triangles for which ''a''
2 > ''b''
2 + ''bc'' + ''c''
2 or ''b''
2 > ''c''
2 + ''ca'' + ''a''
2 or ''c''
2 > ''a''
2 + ''ab'' + ''b''
2.
:*A domain of much practical value since it is dense in T yet excludes all trivial triangles (i.e. points) and degenerate triangles
(i.e. lines) is the set of all
scalene triangles. It is obtained by removing the planes ''b'' = ''c'', ''c'' = ''a'', ''a'' = ''b'' from T.
Domain symmetry
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
equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
triangles.
Examples
Circumcenter
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''(''b''
2 + ''c''
2 − ''a''
2) : ''b''(''c''
2 + ''a''
2 − ''b''
2) : ''c''(''a''
2 + ''b''
2 − ''c''
2).
Let ''f''(''a'',''b'',''c'') = ''a''(''b''
2 + ''c''
2 − ''a''
2). Then
:''f''(''ta'',''tb'',''tc'') = (''ta'') ( (''tb'')
2 + (''tc'')
2 − (''ta'')
2 ) = ''t''
3 ( ''a''( ''b''
2 + ''c''
2 − ''a''
2) ) = ''t''
3 ''f''(''a'',''b'',''c'') (homogeneity)
:''f''(''a'',''c'',''b'') = ''a''(''c''
2 + ''b''
2 − ''a''
2) = ''a''(''b''
2 + ''c''
2 − ''a''
2) = ''f''(''a'',''b'',''c'') (bisymmetry)
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.
1st isogonic center
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 1st isogonal center. Its trilinear coordinates are
:csc(A + π/3) : csc(B + π/3) : csc(C + π/3).
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.
Fermat point
Let
:
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
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...
.
Non-examples
Brocard points
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.
The first and second Brocard points are one of many bicentric pairs of points, 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.
Position vectors
Triangle centers are rendered as
weighted arithmetic mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
and can be written as
:
where
are position vectors of the center (
) and vertices (
), and
are scalars that produce the desired center. Some center instances can be seen in the following table, where
are the lengths of the sides opposite the corresponding vertices, and
is the area of the triangle, as calculated by
Heron's formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is,
:A = \sqrt.
It is named after first-century ...
.
:
:
:
:
Some well-known triangle centers
Classical triangle centers
Recent triangle centers
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.
General classes of triangle centers
Kimberling center
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 called ''Kimberling centers''.
Polynomial triangle center
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''.
Regular triangle center
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.
Major triangle center
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(X) is a function of the angle X alone and does not depend on the other angles or on the side lengths.
Transcendental triangle center
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.
Miscellaneous
Isosceles and equilateral triangles
Let ''f'' be a triangle center function. If two sides of a triangle are equal (say ''a'' = ''b'') then
:
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.
Excenters
Let
:
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
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'')
2 ''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.
New centers from old
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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
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'')
−1(''a''+''b''+''c'')
3''f'' .
Uninteresting centers
Assume ''a'',''b'',''c'' are real variables and let α,β,γ be any three real constants. Let
:
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.
Barycentric coordinates
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 coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
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.
Binary systems
There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed by ''X''
3 and the incenter of the tangential triangle. Consider the triangle center function given by:
:
For the corresponding triangle center there are four distinct possibilities:
:* cos(''A'') : cos(''B'') : cos(''C'') if the reference triangle is acute (this is also the circumcenter).
:*
os(''A'') + sec(''B'')sec(''C''):
os(''B'') − sec(''B''):
os(''C'') − sec(''C'') if the angle at ''A'' is obtuse.
:*
os(''A'') − sec(''A''):
os(''B'') + sec(''C'')sec(''A''):
os(''C'') − sec(''C'') if the angle at ''B'' is obtuse.
:*
os(''A'') − sec(''A''):
os(''B'') − sec(''B''):
os(''C'') + sec(''A'')sec(''B'') if the angle at ''C'' is obtuse.
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
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.
Alternative terminology
Some other names for dilation are
uniform scaling
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a ''scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
,
isotropic scaling,
homothety
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...
, and
homothecy
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...
.
Non-Euclidean and other geometries
The study of triangle centers traditionally is concerned with
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, but triangle centers can also be studied in
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
.
Spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
triangle centers can be defined using
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
. Triangle centers that have the same form for both Euclidean and
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
can be expressed using
gyrotrigonometry
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
.
In non-Euclidean geometry, the assumption that the interior angles of the triangle sum to 180 degrees must be discarded.
Centers of
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
or higher-dimensional
simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
can also be defined, by analogy with 2-dimensional triangles.
Some centers can be extended to polygons with more than three sides. The
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
, for instance, can be found for any polygon. Some research has been done on the centers of polygons with more than three sides.
[{{Cite journal, last1=Prieto-Martínez, first1=Luis Felipe, last2=Sánchez-Cauce, first2=Raquel, date=2021-04-02, title=Generalization of Kimberling's Concept of Triangle Center for Other Polygons, url=https://doi.org/10.1007/s00025-021-01388-4, journal=Results in Mathematics, language=en, volume=76, issue=2, pages=81, doi=10.1007/s00025-021-01388-4, issn=1420-9012, arxiv=2004.01677, s2cid=214795185]
See also
*
Central line
*
Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the ...
*
Triangle conic In triangle geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steine ...
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Central triangle
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Modern triangle geometry
In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and ...
Notes
External links
* Manfred Evers
On Centers and Central Lines of Triangles in the Elliptic Plane* Manfred Evers
On the geometry of a triangle in the elliptic and in the extended hyperbolic plane*
Clark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...
Triangle Centersfrom
University of Evansville
The University of Evansville (UE) is a private university in Evansville, Indiana. It was founded in 1854 as Moores Hill College. The university operates a satellite center, Harlaxton College, in Grantham, England.
UE offers more than 80 differ ...
* Ed Pegg
Triangle Centers in the 2D, 3D, Spherical and Hyperbolicfrom
Wolfram Research
Wolfram Research, Inc. ( ) is an American multinational company that creates computational technology. Wolfram's flagship product is the technical computing program Wolfram Mathematica, first released on June 23, 1988. Other products include Wo ...
.
* Paul Yiu
A Tour of Triangle Geometryfrom
Florida Atlantic University
Florida Atlantic University (Florida Atlantic or FAU) is a Public university, public research university with its main campus in Boca Raton, Florida, and satellite campuses in Dania Beach, Florida, Dania Beach, Davie, Florida, Davie, Fort Lauderd ...
.