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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.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. [15] The incircle radius is no greater than one-ninth the sum of the altitudes. [16]: 289
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. [48] Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. [49] The sum of two side lengths can equal the ...
The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
Bisymmetry in the second and third variables: f(a,b,c) = f(a,c,b). 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 f(a,b,c) : f(b,c,a) : f(c,a,b) is called a triangle center.
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 .
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. [ 1 ] [ 2 ] This statement permits the inclusion of degenerate triangles , but some authors, especially those writing about elementary geometry, will exclude this ...