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Lester's Theorem
In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle. The result is named after June Lester, who published it in 1997, and the circle through these points was called the Lester circle by Clark Kimberling. Lester proved the result by using the properties of complex numbers; subsequent authors have given elementary proofs, proofs using vector arithmetic, and computerized proofs. The center of the Lester circle is also a triangle center. It is the center designated as X(1116) in the Encyclopedia of Triangle Centers. Recently, Peter Moses discovered 21 other 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-dimension ... centers lie on the Lester circle. The points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Lester Theorem
Lester is an ancient Anglo-Saxon surname and given name. People Given name * Lester Bangs (1948–1982), American music critic * Lester Oliver Bankhead (1912–1997), American architect * Lester W. Bentley (1908–1972), American artist from Wisconsin * Lester Bird (1938–2021), second prime minister of Antigua and Barbuda (1994–2004) * Lester D. Boronda (1886–1953), American painter, furniture designer, sculptor * Lester Cotton (born 1996), American football player * Lester del Rey (1915–1993), American science fiction author and editor * Lester Ellis (born 1965), Australian former professional boxer * Lester Flatt (1914–1979), American bluegrass musician * Lester Gillis (1908–1934), better known as Baby Face Nelson, American gangster * Les Gold (born 1950), American pawnbroker and reality TV star * Lester Holt (born 1959), American television journalist * Lester Charles King (1907–1989), English geomorphologist * Lester Lanin (1907–2004), American jazz and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Kiepert Hyperbola
In triangle geometry, the Kiepert conics are two special conics associated with the reference triangle. One of them is a hyperbola, called the Kiepert hyperbola and the other is a parabola, called the Kiepert parabola. The Kiepert conics are defined as follows: :If the three triangles A^\prime BC, AB^\prime C and ABC^\prime, constructed on the sides of a triangle ABC as bases, are similar, isosceles and similarly situated, then the triangles ABC and A^\prime B^\prime C^\prime are in perspective. As the base angle of the isosceles triangles varies between -\pi/2 and \pi/2, the locus of the center of perspectivity of the triangles ABC and A^\prime B^\prime C^\prime is a hyperbola called the Kiepert hyperbola and the envelope of their axis of perspectivity is a parabola called the Kiepert parabola. It has been proved that the Kiepert hyperbola is the hyperbola passing through the vertices, the centroid and the orthocenter of the reference triangle and the Kiepert parabola is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Van Lamoen Circle
In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle T. It contains the circumcenters of the six triangles that are defined inside T by its three medians. Specifically, let A, B, C be the vertices of T, and let G be its centroid (the intersection of its three medians). Let M_a, M_b, and M_c be the midpoints of the sidelines BC, CA, and AB, respectively. It turns out that the circumcenters of the six triangles AGM_c, BGM_c, BGM_a, CGM_a, CGM_b, and AGM_b lie on a common circle, which is the van Lamoen circle of T. History The van Lamoen circle is named after the mathematician who posed it as a problem in 2000. A proof was provided by Kin Y. Li in 2001, and the editors of the Amer. Math. Monthly in 2002. Properties The center of the van Lamoen circle is point X(1153) in Clark Kimberling's comprehensive list of triangle centers. In 2003, Alexey Myakishev and Peter Y. Woo proved that the converse of the theorem is nea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Parry Circle
In geometry, the Parry point is a special point associated with a plane triangle. It is the triangle center designated ''X''(111) in Clark Kimberling's ''Encyclopedia of Triangle Centers''. The Parry point and Parry circle are named in honour of the English geometer Cyril Parry, who studied them in the early 1990s. Parry circle Let be a plane triangle. The circle through the centroid and the two isodynamic points of is called the Parry circle of . The equation of the Parry circle in barycentric coordinates is 3(b^2-c^2)(c^2-a^2)(a^2-b^2)(a^2yz+b^2zx+c^2xy) + (x+y+z)\left( \sum_\text b^2c^2(b^2-c^2)(b^2+c^2-2a^2)x\right) =0 The center of the Parry circle is also a triangle center. It is the center designated as ''X''(351) in the Encyclopedia of Triangle Centers. The trilinear coordinates of the center of the Parry circle are a(b^2-c^2)(b^2+c^2-2a^2) : b(c^2-a^2)(c^2+a^2-2b^2) : c(a^2-b^2)(a^2+b^2-2c^2) Parry point The Parry circle and the circumcircle of triangle i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Neuberg Cubic
In Euclidean geometry, the Neuberg cubic is a special cubic plane curve associated with a reference triangle with several remarkable properties. It is named after Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926), a Luxembourger mathematician, who first introduced the curve in a paper published in 1884. The curve appears as the first item, with identification number K001, in Bernard Gibert's Catalogue of Triangle Cubics which is a compilation of extensive information about more than 1200 triangle cubics. Definitions The Neuberg cubic can be defined as a locus in many different ways. One way is to define it as a locus of a point in the plane of the reference triangle such that, if the reflections of in the sidelines of triangle are , then the lines are concurrent. However, it needs to be proved that the locus so defined is indeed a cubic curve. A second way is to define it as the locus of point such that if are the circumcenters of triangles , then the lines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Fermat Points
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible or, equivalently, the geometric median of the three vertices. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it. The Fermat point gives a solution to the geometric median and Steiner tree problems for three points. Construction The Fermat point of a triangle with largest angle at most 120° is simply its first isogonic center or X(13), which is constructed as follows: # Construct an equilateral triangle on each of two arbitrarily chosen sides of the given triangle. # Draw a line from each new vertex to the opposite vertex of the original triangle. # The two lines intersect at the Fermat point. An alternative method is the following: # On each of two ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Perpendicular Bisector
In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'', a line that passes through the midpoint of a given segment, and the ''angle bisector'', a line that passes through the apex of an angle (that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the ''bisector''. Perpendicular line segment bisector Definition *The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly. *The perpendicular bisector of a line segment AB also has the property that each of its points X is equidistant from segment AB's endpoints: (D)\quad , XA, = , XB, . The proof follows from , MA, =, MB, and Pythagoras' theorem: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^2=, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Parallel (geometry)
In geometry, parallel lines are coplanar infinite straight line (geometry), lines that do not intersecting lines, intersect at any point. Parallel planes are plane (geometry), planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not tangent, touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called ''skew lines''. Line segments and Euclidean vectors are parallel if they have the same direction (geometry), direction or opposite direction (geometry), opposite direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometry, affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Antipodal Points
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center (geometry), center. Given any point on a sphere, its antipodal point is the unique point at greatest distance, whether measured intrinsically (great-circle distance on the surface of the sphere) or extrinsically (Chord (geometry), chordal distance through the sphere's interior). Every great circle on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in spherical geometry depend on choosing non-antipodal points, and degeneracy (mathematics), degenerate if antipodal points are allowed; for example, a sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Rectangular Hyperbola
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciproca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Dao Thanh Oai
The Tao or Dao is the natural way of the universe, primarily as conceived in East Asian philosophy and religion. This seeing of life cannot be grasped as a concept. Rather, it is seen through actual living experience of one's everyday being. The concept is represented by the Chinese character , which has meanings including 'way', 'path', 'road', and sometimes 'doctrine' or 'principle'. In the ''Tao Te Ching'', the ancient philosopher Laozi explains that the Tao is not a name for a thing, but the underlying natural order of the universe whose ultimate essence is difficult to circumscribe because it is non-conceptual yet evident in one's being of aliveness. The Tao is "eternally nameless" and should be distinguished from the countless named things that are considered to be its manifestations, the reality of life before its descriptions of it. Description and uses of the concept The word "Tao" has a variety of meanings in both the ancient and modern Chinese language. Aside fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |