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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 .
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. This resource is hosted at the University of Evansville. It started from a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles by Professor Clark Kimberling. [1]
It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers. [1] [2] For polygons with more than three sides, the incenter only exists for tangential polygons: those that have an incircle that is tangent to each side of the polygon. In this ...
Centroid of a triangle. 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 figure. The same definition extends to any object in -dimensional Euclidean space. [1]
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 f(a,b,c) : f(b,c,a) : f(c,a,b) is called a triangle center. Clark Kimberling is maintaining a website devoted to a compendium of triangle centers.
The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue). The first eight centered triangular numbers on a hex grid
In plane geometry, the Morley centers are two special points associated with a triangle.Both of them are triangle centers.One of them called first Morley center [1] (or simply, the Morley center [2]) is designated as X(356) in Clark Kimberling's Encyclopedia of Triangle Centers, while the other point called second Morley center [1] (or the 1st Morley–Taylor–Marr Center [2]) is designated ...
Lexell's proof by breaking the triangle A ∗ B ∗ C into three isosceles triangles. The main idea in Lexell's c. 1777 geometric proof – also adopted by Eugène Catalan (1843), Robert Allardice (1883), Jacques Hadamard (1901), Antoine Gob (1922), and Hiroshi Maehara (1999) – is to split the triangle into three isosceles triangles with common apex at the circumcenter and then chase angles ...