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Lemoine Point
In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians ( medians reflected at the associated angle bisectors) of a triangle. In other words, it is the isogonal conjugate of the centroid. Ross Honsberger called its existence "one of the crown jewels of modern geometry". In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6).Encyclopedia of Triangle Centers accessed 2014-11-06. For a non-equilateral triangle, it lies in the open punctured at its own center, and could be any point therein. The symmedian point of a triangle with side lengths , and has homogene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemoine Punkt
Lemoine or Le Moine is a French surname meaning "Monk". Notable people with the surname include: * Adolphe Lemoine, known as Lemoine-Montigny (1812–1880), French comic-actor * Anna Le Moine (born 1973), Swedish curler * Antoine Marcel Lemoine (1763–1817) musician, music publisher, father to Henry * Benjamin-Henri Le Moine (1811–1875), Canadian politician and banker * C.W. Lemoine, US author * Claude Lemoine (born 1932), French chess master and journalist * Cyril Lemoine (born 1983), French cyclist * Émile Lemoine (1840–1912), French geometrician * Fabien Lemoine (born 1987), French footballer * Felipe Ribero y Lemoine (1797–1873), Spanish politician, governor, minister and military leader * Gabriel Lemoine (born 2001), Belgian footballer * Henri Lemoine (cyclist) (1909–1981), French cyclist * Henri Lemoine (fraudster) (fl. 1902–1908), French fraudster * Henry Lemoine (1786–1854), piano teacher, music publisher, composer * Jacques-Antoine-Marie Lemoine ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overdetermined System
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent equations, inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others. The terminology can be described in terms of the concept of constraint counting. Each Variable (mathematics), unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint (mathematics), constraint that restricts one degree of freedom. Therefore, the critical case occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint. The ''overdetermined'' case occurs when the syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon Antoine Jean L'Huilier
Simon Antoine Jean L'Huilier (or L'Huillier) (24 April 1750 in Geneva – 28 March 1840 in Geneva) was a Swiss mathematician of French Huguenot descent. He is known for his work in mathematical analysis and topology, and in particular the generalization of Euler's formula for planar graphs. He won the mathematics section prize of the Berlin Academy of Sciences for 1784 in response to a question on the foundations of the calculus. The work was published in his 1787 book ''Exposition elementaire des principes des calculs superieurs''. (A Latin version was published in 1795.) Although L'Huilier won the prize, Joseph Lagrange, who had suggested the question and was the lead judge of the submissions, was disappointed in the work, considering it "the best of a bad lot." Lagrange would go on to publish his own work on foundations. L'Huilier and Cauchy L'Huilier introduced the abbreviation "lim" for limit, using the first three letters of the Latin , with a full stop) to deno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Wilhelm Grebe
Ernst is both a surname and a given name, the German, Dutch, and Scandinavian form of Ernest. Notable people with the name include: Surname * Adolf Ernst (1832–1899) German botanist known by the author abbreviation "Ernst" * Anton Ernst (born 1975), South African film producer * Alice Henson Ernst (1880-1980), American writer and historian * Bastian Ernst (born 1987), German politician * Britta Ernst (born 1961), German politician * Cornelia Ernst (born 1956), German politician * Edzard Ernst (born 1948), German-British academic * Emil Ernst (1889–1942), astronomer * Ernie Ernst (1924/25–2013), American judge * Eugen Ernst (1864–1954), German politician * Fabian Ernst (born 1979), German soccer player * Fedir Ernst (1891-1942), Ukrainian art historian * Gustav Ernst (born 1944), Austrian writer * Heinrich Wilhelm Ernst (1812–1865), Moravian violinist and composer * Jim Ernst (born 1942), Canadian politician * Jimmy Ernst (1920–1984), American painter, son o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Émile Lemoine
Émile Michel Hyacinthe Lemoine (; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most notably, the École Polytechnique. Lemoine taught as a private tutor for a short period after his graduation from the latter school. Lemoine is best known for his proof of the existence of the Lemoine point (or the symmedian point) of a triangle. Other mathematical work includes a system he called ''Géométrographie'' and a method which related algebraic expressions to geometric objects. He has been called a co-founder of modern triangle geometry, as many of its characteristics are present in his work. For most of his life, Lemoine was a professor of mathematics at the École Polytechnique. In later years, he worked as a civil engineer in Paris, and he also took an amateur's interest in music. During his tenure at the École Polytechn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brianchon's Theorem
In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. It is named after Charles Julien Brianchon (1783–1864). Formal statement Let P_1P_2P_3P_4P_5P_6 be a hexagon formed by six tangent lines of a conic section. Then lines \overline,\; \overline,\; \overline (extended diagonals each connecting opposite vertices) intersect at a single point B, the Brianchon point.Whitworth, William Allen. ''Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books Connection to Pascal's theorem The polar reciprocal and projective dual of this theorem give Pascal's theorem. Degenerations As for Pascal's theorem there exist ''degenerations'' for Brianchon's theorem, too: Let coincide two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Triangle
In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus the incircle of the tangential triangle coincides with the circumcircle of the reference triangle. The circumcenter of the tangential triangle is on the reference triangle's Euler line, as is the center of similitude of the tangential triangle and the orthic triangle (whose vertices are at the feet of the altitude (triangle), altitudes of the reference triangle).Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", ''Mathematical Gazette'' 91, November 2007, 436–452. The tangential triangle is homothetic transformation, homothetic to the orthic triangle.Altshiller-Court, Nathan. ''College Geometry'', Dover Publications, 2007 (orig. 1952). A reference triangle and its tangential triangle are in perspective (geometry), persp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Triangle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares Method
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The method is widely used in areas such as regression analysis, curve fitting and data modeling. The least squares method can be categorized into linear and nonlinear forms, depending on the relationship between the model parameters and the observed data. The method was first proposed by Adrien-Marie Legendre in 1805 and further developed by Carl Friedrich Gauss. History Founding The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hesse Normal Form
In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane \mathbb^2, a plane in Euclidean space \mathbb^3, or a hyperplane in higher dimensions.John Vince: ''Geometry for Computer Graphics''. Springer, 2005, , pp. 42, 58, 135, 273 It is primarily used for calculating distances (see point-plane distance and point-line distance). It is written in vector notation as :\vec r \cdot \vec n_0 - d = 0.\, The dot \cdot indicates the dot product (or scalar product). Vector \vec r points from the origin of the coordinate system, ''O'', to any point ''P'' that lies precisely in plane or on line ''E''. The vector \vec n_0 represents the unit normal vector of plane or line ''E''. The distance d \ge 0 is the shortest distance from the origin ''O'' to the plane or line. Derivation/Calculation from the normal form Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
