In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on the placement and scale of the triangle. It is listed as X(11) in Clark Kimberling's Encyclopedia of Triangle Centers, and is named after Karl Wilhelm Feuerbach.
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation
In geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because this problem is first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.
The medial triangle or midpoint triangle of a triangle ABC is the triangle with vertices at the midpoints of the triangle's sides AB, AC and BC. It is the n=3 case of the midpoint polygon of a polygon with n sides. The medial triangle is not the same thing as the median triangle, which is the triangle whose sides have the same lengths as the medians of ABC.
In geometry, a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, i.e. a point that is in the middle of the figure by some measure. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.
In geometry, given a triangle ABC, there exist unique points A´, B´, and C´ on the sides BC, CA, AB respectively, such that:
In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.
In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle ABC is the center of gravity of a homogeneous wire frame in the shape of triangle ABC. The point is named in honor of the 19th-century German geometer Theodor Spieker. The Spieker center is a triangle center and it is listed as the point X(10) in Clark Kimberling's Encyclopedia of Triangle Centers.
In triangle geometry, a Hofstadter point is a special point associated with every plane triangle. In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers. Two of them, the Hofstadter zero-point and Hofstadter one-point, are particularly interesting. They are two transcendental triangle centers. Hofstadter zero-point is the center designated as X(360) and the Hofstafter one-point is the center denoted as X(359) in Clark Kimberling's Encyclopedia of Triangle Centers. The Hofstadter zero-point was discovered by Douglas Hofstadter in 1992.
In triangle geometry, the Apollonius point is a special point associated with a plane triangle. The point is a triangle center and it is designated as X(181) in Clark Kimberling's Encyclopedia of Triangle Centers (ETC). The Apollonius center is also related to the Apollonius problem.
In triangle geometry, the Steiner point is a particular point associated with a plane triangle. It is a triangle center and it is designated as the center X(99) in Clark Kimberling's Encyclopedia of Triangle Centers. Jakob Steiner (1796–1863), Swiss mathematician, described this point in 1826. The point was given Steiner's name by Joseph Neuberg in 1886.
In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986. This is one of the recent triangle centers, having been discovered only in 1986, unlike the classical triangle centers like centroid, incenter, and Steiner point.
In geometry, the isoperimetric point is a special point associated with a plane triangle. The term was originally introduced by G.R. Veldkamp in a paper published in the American Mathematical Monthly in 1985 to denote a point P in the plane of a triangle ABC having the property that the triangles PBC, PCA and PAB have isoperimeters, that is, having the property that
In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and initiated the study of this triangle center in 1987.
In geometry the congruent isoscelizers point is a special point associated with a plane triangle. It is a triangle center and it is listed as X(173) in Clark Kimberling's Encyclopedia of Triangle Centers. This point was introduced to the study of triangle geometry by Peter Yff in 1989.
In geometry the Morley centers are two special points associated with a plane triangle. Both of them are triangle centers. One of them called first Morley center is designated as X(356) in Clark Kimberling's Encyclopedia of Triangle Centers, while the other point called second Morley center is designated as X(357). The two points are also related to Morley's trisector theorem which was discovered by Frank Morley in around 1899.
In geometry the Gossard perspector is a special point associated with a plane triangle. It is a triangle center and it is designated as X(402) in Clark Kimberling's Encyclopedia of Triangle Centers. The point was named Gossard perspector by John Conway in 1998 in honour of Harry Clinton Gossard who discovered its existence in 1916. Later it was learned that the point had appeared in an article by Christopher Zeeman published during 1899 – 1902. From 2003 onwards the Encyclopedia of Triangle Centers has been referring to this point as Zeeman–Gossard perspector.
In geometry central lines are certain special straight lines associated with a plane triangle and lying in the plane of the triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994.
In the geometry of triangles, the Vecten points are two triangle centers associated with any triangle. They may be constructed by constructing three squares on the sides of the triangle, connecting each square centre by a line to the opposite triangle point, and finding the point where these three lines meet. The outer and inner Vecten points differ according to whether the squares are extended outward from the triangle sides, or inward.
In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle . It contains the circumcenters of the six triangles that are defined inside by its three medians.
