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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 intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GEOS Circle Overview
GEOS may refer to: Computer software * GEOS (8-bit operating system), an operating system originally designed for the Commodore 64 * GEOS (16-bit operating system), a DOS-based graphical user interface and x86 operating system * GEOS (securities processing software), an integrated online system for the management and processing of securities * GEOS (software library), an open-source geometry engine * Goddard Earth Observing System, an Earth system model Other * GEOS (eikaiwa), an English conversation teaching company based in Japan * GEOS circle, an intersection of four lines that are associated with a generalized triangle * GEOS (satellite), a research satellite from ESRO (1978–1982) * GEOS (satellite series), three research satellites from NASA * Groupe GEOS, a French business consultancy {{disambiguation ... [...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 intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...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 ... [...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 Soddy line intersects the Euler line in the de Longchamps point and 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 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 in 1936. References * Zuming Feng: ''Why Are the Gergonne and Soddy Lines Perpendicular? A Synthetic Approach''. In: ''Mathematics Magazin'', Band 81, Nr. 3, Juni 2008, S. 211-214JSTOR *Roger Alperin: ''The Gergonne and Soddy lines''. In: ''Elemente der Mathematik ''Elemente der Mathematik'' is a peer-reviewed scientific journal covering mathematics. It is published by the Europea ... [...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 jour ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically used for lines and planes that intersect to form a right angle, whereas ''orthogonal'' is used in generalizations, such as ''orthogonal vectors'' or ''orthogonal curves''. ''Orthogonality'' is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. 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 Optics In optics, polarization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocentric System
In geometry, an orthocentric system is a set (mathematics), set of four point (geometry), points on a plane (mathematics), 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 c ... [...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-po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fletcher Point
Fletcher may refer to: People and fictional characters * Fletcher (surname), including lists of people and fictional characters * Fletcher (given name), lists of people and fictional characters * Fletcher (occupation), a person who fletches arrows, the origin of the surname * Fletcher (singer), American singer-songwriter Cari Fletcher (born 1994) 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, a mountain range * Fletcher Pond, Michigan, a man-made body ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
