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Central Line (geometry)
In geometry, central lines are certain special straight lines that lie in the Plane (geometry), 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 whose equation in trilinear coordinates has the form f(a,b,c)\,x + g(a,b,c)\,y + h(a,b,c)\,z = 0 where the point with trilinear coordinates f(a,b,c) : g(a,b,c) : h(a,b,c) is a triangle center, is a central line in the plane of relative to . 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 pola ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incentral 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] |
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 Euler line of its triangle, at the |
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
The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute triangle, acute. For a right triangle, the orthocenter coincides with the vertex (geometry), vertex at the right angle. For an equilateral triangle, all triangle center, triangle centers (including the orthocenter) coincide at its centroid. Formulation Let denote the vertices and also the angles of the triangle, and let a = \left, \overline\, b = \left, \overline\, c = \left, \overline\ be the side lengths. The orthocenter has trilinear coordinatesClark Kimberling's Encyclopedia of Triangle Centers \begin & \sec A:\sec B:\sec C \\ &= \cos A-\sin B \sin C:\cos B-\sin C \sin A:\cos C-\sin A\sin B, \end and Barycentric coordinates (mathematics), barycentric coordinates \begin & (a^2+b^2-c^2)(a^2-b^2+c^2) : (a^2+b^2-c^2)(-a^2+b^2+c^2) : (a^2- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcenter
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case , a cyclic quadrilateral. All rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is. Straightedge and compass construction The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. For three non-collinear points, these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector is equidistant from th ... [...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] |
Symmedian Point
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry".. Isogonality Many times in geometry, if we take three special lines through the vertices of a triangle, or '' cevians'', then their reflections about the corresponding angle bisectors, called ''isogonal lines'', will also have interesting properties. For instance, if three cevians of a triangle inters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-dimensional Euclidean space. In geometry, one often assumes uniform mass density, in which case the '' barycenter'' or ''center of mass'' coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin. In physics, if variations in gravity are considered, then a '' center of gravity'' can be defined as the weighted mean of all points weighted by their specific weight. In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center. History The term "centroid" was coined in 1814. It is used as a substitute for the older terms "center of grav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemoine 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 whose equation in trilinear coordinates has the form f(a,b,c)\,x + g(a,b,c)\,y + h(a,b,c)\,z = 0 where the point with trilinear coordinates f(a,b,c) : g(a,b,c) : h(a,b,c) is a triangle center, is a central line in the plane of relative to . 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 co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
