In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. 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. 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. , Clark Kimberling's Encyclopedia of Triangle Centers 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 of a triangle is the second entry and is denoted by or .
Formal definition
A real-valued function of three real variables may have the following properties: Homogeneity: for some constant and for all . Bisymmetry in the second and third variables: If a non-zero has both these properties it is called a triangle center function. If is a triangle center function and 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 . This process is known as cyclicity.
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 .
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 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 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 . For example:
The centers X3, X4, X22, X24, X40 make specific reference to acute triangles, namely that region of 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 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 , , from .