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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] |
Inellipse
In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, the Mandart inellipse and Brocard inellipse (see examples section). For any triangle there exist an infinite number of inellipses. The Steiner inellipse plays a special role: Its area is the greatest of all inellipses. Because a non-degenerate conic section is uniquely determined by five items out of the sets of vertices and tangents, in a triangle whose three sides are given as tangents one can specify only the points of contact on two sides. The third point of contact is then uniquely determined. Parametric representations, center, conjugate diameters The inellipse of the triangle with vertices :O=(0,0), \; A=(a_1,a_2), \; B=(b_1,b_2) and points of contact :U=(u_1,u_2) ,\; V=(v_1,v_2) on OA and OB respectively can by describe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on a Euclidean plane. 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 thous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Projective Geometry
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conic Sections
A conic section, conic or a quadratic curve is a curve obtained from a Conical surface, cone's surface intersecting a plane (mathematics), 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 it was sometimes considered a fourth type. The Greek mathematics, 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 (mathematics), set of those points whose distances to some particular point, called a ''Focus (geometry), focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''Eccentricity (mathematics), eccentricity''. The type of conic is determined by the value of the ec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seven Circles Theorem
In geometry, the seven circles theorem is a theorem about a certain arrangement of seven circles in the Euclidean plane. Specifically, given a chain of six circles all tangent to a seventh circle and each tangent to its two neighbors, the three lines drawn between opposite pairs of the points of tangency on the seventh circle all pass through the same point. Though elementary in nature, this theorem was not discovered until 1974 (by Evelyn, Money-Coutts, and Tyrrell). See also * Brianchon's theorem References * * * External links * {{MathWorld, title=Seven Circles Theorem, urlname=SevenCirclesTheorem Interactive Appletby Michael Borcherds showing The Seven Circles Theorem made usinGeoGebra the Seven Circles Theoremby Stanley Rabinowitz, with a proof based on Ceva's theorems. Seven Circles Theoremat Cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Axis
In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail: For two circles with centers and radii the powers of a point with respect to the circles are :\Pi_1(P)=, PM_1, ^2 - r_1^2,\qquad \Pi_2(P)= , PM_2, ^2 - r_2^2. Point belongs to the radical axis, if : \Pi_1(P)=\Pi_2(P). If the circles have two points in common, the radical axis is the common secant line of the circles. If point is outside the circles, has equal tangential distance to both the circles. If the radii are equal, the radical axis is the line segment bisector of . In any case the radical axis is a line perpendicular to \overline. ;On notations The notation ''radical axis'' was used by the French mathematician Michel Chasles, M. Chasles as ''axe radical''. Jean-Victor Poncelet, J.V. Poncelet us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line At Infinity
In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. Geometric formulation In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel. Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept. In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a Point (geometry), point (the Focus (geometry), focus) and a Line (geometry), line (the Directrix (conic section), directrix). The focus does not lie on the directrix. The parabola is the locus (mathematics), locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane (geometry), plane Parallel (geometry), parallel to another plane that is tangential to the conical surface. The graph of a function, graph of a quadratic function y=ax^2+bx+ c (with a\neq 0 ) is a parabola with its axis parallel to the -axis. Conversely, every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Real Projective Plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the setting for planar projective geometry, in which the relationships between objects are not considered to change under projective transformations. The name ''projective'' comes from perspective drawing: projecting an image from one plane onto another as viewed from a point outside either plane, for example by photographing a flat painting from an oblique angle, is a projective transformation. The fundamental objects in the projective plane are points and straight lines, and as in Euclidean geometry, every pair of points determines a unique line passing through both, but unlike in the Euclidean case in projective geometry every pair of lines also determines a unique point at their intersection (in Euclidean geometry, parallel lines never in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of each point (mathematics), point. It is an affine space, which includes in particular the concept of parallel lines. It has also measurement, metrical properties induced by a Euclidean distance, distance, which allows to define circles, and angle, angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a ''Cartesian plane''. The set \mathbb^2 of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called ''the'' Euclidean plane or ''standard Euclidean plane'', since every Euclidean plane is isomorphic to it. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
