Triangle center

For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.

Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations.

This invariance is the defining property of a triangle center.

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.

[3] Every entry in the Encyclopedia of Triangle Centers is denoted by

⁠ For example, the trilinears of X365 which is the 365th entry in the Encyclopedia of Triangle Centers, are

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

In order to support the bisymmetry test D must be symmetric about the planes b = c, c = a, a = b.

The point of concurrence of the perpendicular bisectors of the sides of triangle △ABC is the circumcenter.

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.

These coordinates satisfy the properties of homogeneity and cyclicity but not bisymmetry.

So the first Brocard point is not (in general) a triangle center.

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.

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

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.

If such a function is also non-zero and homogeneous it is easily seen that the mapping

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.

Another system is formed by X3 and the incenter of the tangential triangle.

For the corresponding triangle center there are four distinct possibilities:

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.

In the image the coordinates refer to the (c, b, a) triangle and (using "|" as the separator) the reflection of an arbitrary point

Since rotations and translations may be regarded as double reflections they too must preserve triangle centers.

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.