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Droz-Farny Line Theorem
In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle. Let T be a triangle with vertices A, B, and C, and let H be its orthocenter (the common point of its three altitude lines. Let L_1 and L_2 be any two mutually perpendicular lines through H. Let A_1, B_1, and C_1 be the points where L_1 intersects the side lines BC, CA, and AB, respectively. Similarly, let Let A_2, B_2, and C_2 be the points where L_2 intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments A_1A_2, B_1B_2, and C_1C_2 are collinear. The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof. Goormaghtigh's generalization A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh. As above, let T be a triangle with vertices A, B, and C. Let P be any point distinct from A, B, and C, and L be any line through ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conic
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a '' focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the '' eccentricity''. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-the-Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games. He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website ''Cut-the-Knot'' for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started ''Cut-the-Knot'' in October 1996.Interview with Alexander ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathworld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. Weisstein continuously improved the notes and accepted corrections and comments from online readers. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pole And Polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole. Properties Pole and polar have several useful properties: * If a point P lies on the line ''l'', then the pole L of the line ''l'' lies on the polar ''p'' of point P. * If a point P moves along a line ''l'', its polar ''p'' rotates about the pole L of the line ''l''. * If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points. * If a point lies on the conic section, its polar is the tangent through this point to the conic section. * If a point P lies on its own polar line, then P is on the conic section. * Each line has, with respect to a non-degenerated conic section, exactly one pole. Special case of circles The pole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane (geometry)
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane. Euclidean geometry Euclid set forth the first great landmark of mathematical thought, an axio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point (geometry)
In classical Euclidean geometry, a point is a primitive notion that models an exact location in space, and has no length, width, or thickness. In modern mathematics, a point refers more generally to an element of some set called a space. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy; for example, ''"there is exactly one line that passes through two different points"''. Points in Euclidean geometry Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as "that which has no part". In two-dimensional Euclidean space, a point is represented by an ordered pair (, ) of numbers, where the first number conventionally represents the horizontal and is often denoted by , and the second number conventionally represents the vertical and is often denoted b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dao Thanh Oai
Dao, Dão or DAO may refer to: * Tao (Chinese: "The Way" 道), a philosophical concept * Dao (Chinese sword) (刀), a type of Chinese sword * Dao (Naga sword), a weapon and a tool of Naga people People and language * Yao people, a minority ethnic group of Vietnam * Dao language (Papuan), Indonesia * Dao language (China) * Dao (surname) (Đào), a Vietnamese surname * Dao (''Dungeons & Dragons''), a type of genie in the game ''Dungeons & Dragons'' * Dão (footballer) (born 1984), Brazilian football defender Places * Dao (country subdivision) (Dào), historical political divisions in China translated as "circuits" * Dao (state), a historical state during the Zhou dynasty * Dao, Capiz, Philippines * Dao County, in Yongzhou, Hunan, China * Dão DOC, a wine region in Portugal * Dão River, a river in Portugal Science and technology Biology * D-amino acid oxidase, a peroxisomal enzyme * Diamine oxidase, an enzyme also known as histaminase involved in the metabolism of histamine * D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Goormaghtigh
René Goormaghtigh (13 October 1893, Ostend – 10 February 1960, Ixelles) was a Belgian engineer, after whom the Goormaghtigh Conjecture is named. Goormaghtigh studied at Ghent University, gaining a Diploma in Civil Engineering from the Central Board of Le Havre in 1918. Throughout his subsequent life he worked as an engineer and industrial administrator. In 1952 he was appointed advisor to the Société Générale de Belgique. He was made a Knight of the Order of Leopold II The Order of Leopold II is an order of Belgium and is named in honor of King Leopold II. The decoration was established on 24 August 1900 by Leopold II as Sovereign of the Congo Free State and was in 1908, upon Congo being handed over to Belgium ... in 1947, and an Officer of the Order of the Crown in 1956. After a heart attack in 1958, he retired to Saint-André-des-Bruges.Fransisco Bellot-RosadoRené Gormaghtigh, ingénieur et géomètre de MATHESIS/ref> He was a frequent contributor to mathematical jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
