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Triangle center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a ''center'' of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of them has the property that it is invariant under similarity. In other words, it will always occupy the same position (relative to the vertices) under the operations of rotation, reflection, and dilation. Consequently, this invariance is a necessary property for any point being considered as a triangle center. It rules out various well-known points such as the Brocard points, named after Henri Brocard (1845–1922), which are not invariant under reflection and so fail to qualify as 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, symmedian 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.〔List of classical and recent triangle centers: (【引用サイトリンク】title=Triangle centers )〕〔Summary of ''Central Points and Central Lines in the Plane of a Triangle '' () (Accessed on 23 may 2009)〕 , Clark Kimberling's Encyclopedia of Triangle Centers contains an annotated list of 6,102 triangle centers.〔(Centers X(5001) - )〕
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