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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 ...
A triangle center P is said to be a major triangle center if the trilinear coordinates of P can be expressed in the form (): (): where is a function of the angle X alone and does not depend on the other angles or on the side lengths. [9]
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]
The Nagel point is the isotomic conjugate of the Gergonne point.The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line.The incenter is the Nagel point of the medial triangle; [2] [3] equivalently, the Nagel point is the incenter of the anticomplementary triangle.
The triangle DEF is called the pedal triangle of P. [17] The antipedal triangle of P is the triangle formed by the lines through A, B, C perpendicular to PA, PB, PC respectively. Two points P and Q are called counter points if the pedal triangle of P is homothetic to the antipedal triangle of Q and the pedal triangle of Q is homothetic to the ...
Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point. In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible [1] or ...
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 ...
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