|
GEOS Circle
In geometry, the GEOS circle is derived from the intersection of four lines that are associated with a generalized triangle: the Euler line, the Soddy line, the orthic axis and the Gergonne line. Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is orthogonal to the Gergonne line. These four lines provide six points of intersection of which two points occur at line intersections that are orthogonal. Consequently, the other four points form an orthocentric system. The GEOS circle is that circle centered at a point equidistant from ''X''650 (the intersection of the orthic axis with the Gergonne line) and ''X''20 (the intersection of the Euler line with the Soddy line and is known as the de Longchamps point) and passes through these points as well as the two points of orthogonal intersection. The orthogonal intersection points are ''X''468 (the intersection of the orthic axis with the Euler line) and ''X''1323 (the Fletcher point, the intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GEOS Circle Overview
{{redirect category shell, {{R from other capitalisation{{R from ambiguous page ...
#REDIRECT GEOS #REDIRECT GEOS {{redirect category shell, {{R from other capitalisation{{R from ambiguous page ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GEOS Circle
In geometry, the GEOS circle is derived from the intersection of four lines that are associated with a generalized triangle: the Euler line, the Soddy line, the orthic axis and the Gergonne line. Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is orthogonal to the Gergonne line. These four lines provide six points of intersection of which two points occur at line intersections that are orthogonal. Consequently, the other four points form an orthocentric system. The GEOS circle is that circle centered at a point equidistant from ''X''650 (the intersection of the orthic axis with the Gergonne line) and ''X''20 (the intersection of the Euler line with the Soddy line and is known as the de Longchamps point) and passes through these points as well as the two points of orthogonal intersection. The orthogonal intersection points are ''X''468 (the intersection of the orthic axis with the Euler line) and ''X''1323 (the Fletcher point, the intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Line
In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. Triangle centers on the Euler line Individual centers Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. This property is also true for another triangle center, the nine-point center, although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them. Other notable points that lie on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soddy Line
The Soddy line of a triangle is the line that goes through the centers of the two Soddy circles of that triangle. The Soody line intersects the Euler line in de Longchamps point und the Gergonne line in the ''Fletcher point''. It is also perpendicular to the Gergonne line and together all three lines form the Euler-Gergonne-Soddy triangle. The Gergonne point and the incenter of the triangle are located on the on the Soddy line as well. The line is named after Nobel laureate Frederick Soddy, who published a proof of a special case of Descartes' theorem about tangent circles as a poem in Nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ... in 1936. References * Zuming Feng: ''Why Are the Gergonne and Soddy Lines Perpendicular? A Synthetic Approach''. In: ''Mathematics Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthic Axis
In geometry, central lines are certain special straight lines that lie in the plane of a 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. Definition Let be a plane triangle and let be the trilinear coordinates of an arbitrary point in the plane of triangle . A straight line in the plane of triangle whose equation in trilinear coordinates has the form : where the point with trilinear coordinates is a triangle center, is a central line in the plane of triangle relative to the triangle . Central lines as trilinear polars The geometric relation between a central line and its associated triangle center can be expressed using the concepts of trilinear polars and isogonal conjugates. Let ''X'' = ( ''u'' ( ''a'', ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gergonne Line
Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician. Life In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion because the French government feared a foreign invasion intended to undo the French Revolution and restore Louis XVI to the throne of France. He saw action in the major battle of Valmy on 20 September 1792. He then returned to civilian life but soon was called up again and took part in the French invasion of Spain in 1794. In 1795, Gergonne and his regiment were sent to Nîmes. At this point, he made a definitive transition to civilian life by taking up the chair of "transcendental mathematics" at the new École centrale. He came under the influence of Gaspard Monge, the Director of the new École polytechnique in Paris. In 1810, in response to difficulties he encountered in trying to publish his work, Gergonne founded his own mathematics journ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocentric System
In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and the four circles passing through any three of the four points have the same radius. If four points form an orthocentric system, then ''each'' of the four points is the orthocenter of the other three. These four possible triangles will all have the same nine-point circle. Consequently these four possible triangles must all have circumcircles with the same circumradius. The common nine-point circle The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the common nine-point circle passes. The nine-point circle al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Longchamps Point
In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.. Definition Let the given triangle have vertices A, B, and C, opposite the respective sides a, b, and c, as is the standard notation in triangle geometry. In the 1886 paper in which he introduced this point, de Longchamps initially defined it as the center of a circle \Delta orthogonal to the three circles \Delta_a, \Delta_b, and \Delta_c, where \Delta_a is centered at A with radius a and the other two circles are defined symmetrically. De Longchamps then also showed that the same point, now known as the de Longchamps point, may be equivalently defined as the orthocenter of the anticomplementary triangle of ABC, and that it is the reflection of the orthocenter of ABC around the circumcenter. The Steiner circle of a triangle is concentric with the nine-point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fletcher Point
Fletcher may refer to: People * Fletcher (occupation), a person who fletches arrows, the origin of the surname * Fletcher (singer) (born 1994), American actress and singer-songwriter * Fletcher (surname) * Fletcher (given name) Places United States * Fletcher, California, a former settlement * Fletcher, the original name of Aurora, Colorado, a home rule municipality * Fletcher, Illinois, an unincorporated community * Fletcher, Indiana, an unincorporated town * Fletcher, Missouri, an unincorporated community * Fletcher, North Carolina, a suburb of Asheville * Fletcher, Ohio, a village * Fletcher, Oklahoma, a town * Fletcher, Vermont, a town * Fletcher, Virginia, an unincorporated community * Fletcher, West Virginia, an unincorporated community * Fletcher Hills, San Diego County, California * Fletcher Pond, Michigan, a man-made body of water Antarctica * Fletcher Islands, George V Land * Fletcher Island, largest of the Fletcher Islands * Fletcher Peninsula, Ellsworth La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
