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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 n-th centered triangular number, corresponding to n layers plus the center, is given by the formula:, = + (+) = + +. Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number.
The center of the incircle is a triangle center called the triangle's incenter. [1] An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. [3]
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 surface of the figure. [further explanation needed] The same definition extends to any object in -dimensional Euclidean space. [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.
Hence, given the radius, r, center, P c, a point on the circle, P 0 and a unit normal of the plane containing the circle, ^, one parametric equation of the circle starting from the point P 0 and proceeding in a positively oriented (i.e., right-handed) sense about ^ is the following:
The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane.
A straight line in the plane of triangle ABC whose equation in trilinear coordinates has the form f ( a, b, c) x + g ( a, b, c) y + h ( a, b, c) z = 0. where the point with trilinear coordinates ( f ( a, b, c) : g ( a, b, c) : h ( a, b, c) ) is a triangle center, is a central line in the plane of triangle ABC relative to the triangle ABC. [25] [26]