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Central Line (geometry)
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994. Definition Let be a plane triangle and let be the trilinear coordinates of an arbitrary point in the plane of triangle . A straight line in the plane of triangle whose equation in trilinear coordinates has the form : where the point with trilinear coordinates is a triangle center, is a central line in the plane of triangle relative to the triangle . Central lines as trilinear polars The geometric relation between a central line and its associated triangle center can be expressed using the concepts of trilinear polars and isogonal conjugates. Let ''X'' = ( ''u'' ( ''a'', ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemoine Axis
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 * 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 (1751–1824), also Lemoyne, French painter * Jake Lemoine (born 1993), American baseball player * James MacPherson Le Moine (1825–1912), Canadian writer, lawyer and historian * Jean Lemoine (1250–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spieker Center
In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle is the center of gravity of a homogeneous wire frame in the shape of . The point is named in honor of the 19th-century German geometer Theodor Spieker. The Spieker center is a triangle center and it is listed as the point ''X''(10) in Clark Kimberling's Encyclopedia of Triangle Centers. Location The following result can be used to locate the Spieker center of any triangle. :The Spieker center of triangle is the incenter of the medial triangle of . That is, the Spieker center of is the center of the circle inscribed in the medial triangle of . This circle is known as the Spieker circle. The Spieker center is also located at the intersection of the three cleavers of triangle . A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 lie on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Medial Triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is not the same thing as the median triangle, which is the triangle whose sides have the same lengths as the medians of . Each side of the medial triangle is called a ''midsegment'' (or ''midline''). In general, a midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to half the length of the third side. Properties The medial triangle can also be viewed as the image of triangle transformed by a homothety centered at the centroid with ratio -1/2. Thus, the sides of the medial triangle are half and parallel to the corresponding sides of triangle ABC. Hence, the medial triangle is inversely similar and shares the same centroid and medians with ... [...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 real (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 intersect at a point on the line at infinity, 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 op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nine-point Center
In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices. The nine-point center is listed as point X(5) in Clark Kimberling's Encyclopedia of Triangle Centers..Encyclopedia of Triangle Centers accessed 2014-10-23. Properties The nine-point center lies on the of its triangle, a ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Line Of Orhocenter
Central is an adjective usually referring to being in the center of some place or (mathematical) object. Central may also refer to: Directions and generalised locations * Central Africa, a region in the centre of Africa continent, also known as Middle Africa * Central America, a region in the centre of America continent * Central Asia, a region in the centre of Eurasian continent * Central Australia, a region of the Australian continent * Central Belt, an area in the centre of Scotland * Central Europe, a region of the European continent * Central London, the centre of London * Central Region (other) * Central United States, a region of the United States of America Specific locations Countries * Central African Republic, a country in Africa States and provinces * Blue Nile (state) or Central, a state in Sudan * Central Department, Paraguay * Central Province (Kenya) * Central Province (Papua New Guinea) * Central Province (Solomon Islands) * Central Province, Sri Lank ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ''extended base'' of the altitude. The intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an obtuse triangle, the minimum bounding circle has the longest side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthic Axis
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994. Definition Let be a plane triangle and let be the trilinear coordinates of an arbitrary point in the plane of triangle . A straight line in the plane of triangle whose equation in trilinear coordinates has the form : where the point with trilinear coordinates is a triangle center, is a central line in the plane of triangle relative to the triangle . Central lines as trilinear polars The geometric relation between a central line and its associated triangle center can be expressed using the concepts of trilinear polars and isogonal conjugates. Let ''X'' = ( ''u'' ( ''a'', ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
