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Pascal Theorem
In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'', Latin for mystical hexagram) states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon ( extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal. The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel. This theorem is a generalization of Pappus's (hexagon) theorem, which is the special case of a degenerate conic of two lines with three points on each line. Euclidean variants The most natural setting for Pascal's theorem is in a projective plane since any two lines meet and no exceptions need to be made for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in Perspective (graphical)#Renaissance, perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Braikenridge–Maclaurin Construction
In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve). There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines. Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non- degenerate; this is true over both the Euclidean plane and any pappian projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion. Proofs This result can be proven numerous different ways; the dimension counting argument is most direct, and generalizes to higher degree, while other proofs are special to coni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Colin Maclaurin
Colin Maclaurin (; ; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for being the youngest professor. The Maclaurin series, a special case of the Taylor series, is named after him. Owing to changes in orthography since that time (his name was originally rendered as M'Laurine), his surname is alternatively written MacLaurin. Early life Maclaurin was born in Kilmodan, Argyll. His father, John Maclaurin, minister of Glendaruel, died when Maclaurin was in infancy, and his mother died before he reached nine years of age. He was then educated under the care of his uncle, Daniel Maclaurin, minister of Kilfinan. A child prodigy, he entered university at age 11. Academic career At eleven, Maclaurin, a child prodigy at the time, entered the University of Glasgow. He graduated Master of Arts three years later by defending a thesis on ''the Power ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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William Braikenridge
William Braikenridge (also Brakenridge) (c.1700–1762) was a Scottish mathematician and cleric, a Fellow of the Royal Society from 1752. Life He was son of John Braikenridge of Glasgow. s:Page:Alumni Oxoniensis (1715–1886) volume 1.djvu/169 In the 1720s he taught mathematics in Edinburgh. Braikenridge was Honorary A.M. in 1735, and D.D. in 1739, of Marischal College, when he was vicar of New Church, Isle of Wight. He was incorporated at The Queen's College, Oxford, in 1741. He became rector of St Michael Bassishaw, and from 1745 librarian of Sion College, in London. Works In geometry the Braikenridge–Maclaurin theorem was independently discovered by Colin Maclaurin. It occasioned a priority dispute after Braikenridge published it in 1733; Stella Mills writes that, while Braikenridge may have wished to establish priority, Maclaurin rather felt slighted by the implication that he did not know theorems in the ''Exercitatio'' that he had taught for a number of years. *'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Braikenridge–Maclaurin Theorem
In geometry, the , named for 18th-century British mathematicians William Braikenridge and Colin Maclaurin, is the converse to Pascal's theorem. It states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line ''L'', then the six vertices of the hexagon lie on a conic ''C''; the conic may be degenerate, as in Pappus's hexagon theorem. The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a synthetic Synthetic may refer to: Science * Synthetic biology * Synthetic chemical or compound, produced by the process of chemical synthesis * Synthetic elements, chemical elements that are not naturally found on Earth and therefore have to be created in ... construction of the conic defined by five points, by varying the sixth point. Namely, Pascal's theorem states that given six points on a conic (the vertices of a hexagon), the lines defined by opposite sides intersect in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Five Points Determine A Conic
In Euclidean geometry, Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) Point (geometry), points determine a line (geometry), line (a degree-1 plane curve). There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines. Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-Degenerate conic, degenerate; this is true over both the Euclidean plane and any Pappian plane, pappian projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see Degenerate conic#Points to define, further discussion. Proofs This result can be proven numerous different ways; the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gergonne 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. (La Hire's theorem) * 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cayley–Bacharach Theorem
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states: :Assume that two cubics and in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point. A more intrinsic form of the Cayley–Bacharach theorem reads as follows: :Every cubic curve over an algebraically closed field that passes through a given set of eight points also passes through (counting multiplicities) a ninth point which depends only on . A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and Isaak Bacharach. Details If seven of the points lie on a conic, then the ninth point can be chosen on that conic, since will always contain the whole conic on account of Bézout's the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Broadside (printing)
A broadside is a large sheet of paper printed on one side only. Historically in Europe, broadsides were used as posters, announcing events or proclamations, giving political views, commentary in the form of broadside ballad, ballads, or simply advertisements. In Japan, Chromoxylography, chromoxylographic broadsheets featuring artistic prints were common. Description and history The historical type of broadsides, designed to be plastered onto walls as a form of street literature, were ephemera, i.e., temporary documents created for a specific purpose and intended to be thrown away. They were one of the most common forms of printed material between the sixteenth and nineteenth centuries. They were often advertisements, but could also be used for news information or proclamations. Broadsides were a very popular medium for printing topical ballads starting in the 16th century. Broadside (music), Broadside ballads were usually printed on the cheapest type of paper available. Initiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |