Each of these classical centers has the property that it is
invariant (more precisely
equivariant) under
similarity transformations. In other words, for any triangle and any similarity transformation (such as a
rotation,
reflection,
dilation, or
translation), 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 which are not invariant under reflection and so fail to qualify as triangle centers.
For an
equilateral 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 thousands of triangle centers have been collected in the Encyclopedia of Triangle Centers.
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,
nine-point center,
Lemoine 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.[1][2]Clark Kimberling's Encyclopedia of Triangle Centers contains an annotated list of over 50,000 triangle centers.[3] Every entry in the Encyclopedia of Triangle Centers is denoted by or where is the positional index of the entry. For example, the
centroid of a triangle is the second entry and is denoted by or .
Formal definition
A
real-valued functionf of three real variables a, b, c may have the following properties:
Homogeneity: for some constant n and for all t > 0.
Bisymmetry in the second and third variables:
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 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 of a, b, c. This process is known as cyclicity.[4][5]
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 and 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, 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 be 0 if and 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 For example, the trilinears of X365 which is the 365th entry in the
Encyclopedia of Triangle Centers, are 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 is restricted to the region of where
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 X3, X4, X22, X24, X40 make specific reference to
acute triangles, namely that region of T where
When differentiating between the Fermat point and X13 the domain of triangles with an angle exceeding 2π/3 is important; in other words, triangles for which any of the following is true:
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 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
Let It can be shown that f is homogeneous:
as well as bisymmetric:
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', CC' are concurrent and the point of concurrence is the 1st isogonal center. Its trilinear coordinates are
Expressing these coordinates in terms of a, 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
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.
The trilinear coordinates of the first Brocard point are:
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:
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.
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.
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.
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.[8]
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, 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 △, a, b, c, where △ 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 where is a function of the angle X alone and does not depend on the other angles or on the side lengths.[9]
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, 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
If such a function is also non-zero and homogeneous it is easily seen that the mapping
is a triangle center function. The corresponding triangle center is
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
Together with the zero function, normalized triangle center functions form an
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
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
Since these are precisely the
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.
Binary systems
There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed by X3 and the incenter of the tangential triangle. Consider the triangle center function given by:
For the corresponding triangle center there are four distinct possibilities:
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
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 is If f is a triangle center function the reflection of its triangle center is which, by bisymmetry, is the same as
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.
The study of triangle centers traditionally is concerned with
Euclidean geometry, but triangle centers can also be studied in
non-Euclidean geometry.[10] Triangle centers that have the same form for both Euclidean and
hyperbolic geometry can be expressed using
gyrotrigonometry.[11][12][13] In non-Euclidean geometry, the assumption that the interior angles of the triangle sum to 180 degrees must be discarded.
Centers of
tetrahedra or higher-dimensional
simplices can also be defined, by analogy with 2-dimensional triangles.[13]
Some centers can be extended to polygons with more than three sides. The
centroid, 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]
^actually the 1st isogonic center, but also the Fermat point whenever A,B,C ≤ 2π/3
^Kimberling, Clark.
"Triangle centers". Retrieved 2009-05-23. 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
^Weisstein, Eric W.
"Kimberling Center". MathWorld–A Wolfram Web Resource. Retrieved 25 May 2009.
^Weisstein, Eric W.
"Major Triangle Center". MathWorld–A Wolfram Web Resource. Retrieved 25 May 2009.
^Russell, Robert A. (2019-04-18). "Non-Euclidean Triangle Centers".
arXiv:1608.08190 [
math.MG].
^Ungar, Abraham A. (2009).
"Hyperbolic Barycentric Coordinates"(PDF). The Australian Journal of Mathematical Analysis and Applications. 6 (1): 1–35., article #18