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Central Triangle
In geometry, a central triangle is a triangle in the plane of the reference triangle. The trilinear coordinates of its vertices relative to the reference triangle are expressible in a certain cyclical way in terms of two functions having the same degree of homogeneity. At least one of the two functions must be a triangle center function. The excentral triangle is an example of a central triangle. The central triangles have been classified into three types based on the properties of the two functions. Definition Triangle center function A triangle center function is a real valued function of three real variables having the following properties: :* Homogeneity property: F(tu,tv,tw) = t^n F(u,v,w) for some constant and for all . The constant is the degree of homogeneity of the function :*Bisymmetry property: F(u,v,w) = F(u,w,v). Central triangles of Type 1 Let and be two triangle center functions, not both identically zero functions, having the same degree of homog ... [...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] |
Trilinear Coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point (geometry), point where two or more curves, line (geometry), lines, or edge (geometry), edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedron, polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a vertex is called "convex set, convex" if the internal an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''degree''; that is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain and codomain are vector spaces over a field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of such that \mathbf\in C implies s\mathbf\in C for every nonzero scalar . In the case of functions of several real variables and real vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Center Function
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations. In other words, for any triangle and any similarity transformation (such as a rotation, reflection, dilation, or translation), the center of the transformed triangle is the same point as the transformed center of the original triangle. This invariance is the defining property of a triangle center. It rules out other well-known points such as the Brocard points which are not invariant under reflection and so fail to qualify as triangle centers. For an equilateral triangle, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excentral 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 extended side, 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 and external angle, internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the Internal and external angle, 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 perp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations. In other words, for any triangle and any similarity transformation (such as a rotation, reflection, dilation, or translation), the center of the transformed triangle is the same point as the transformed center of the original triangle. This invariance is the defining property of a triangle center. It rules out other well-known points such as the Brocard points which are not invariant under reflection and so fail to qualify as triangle centers. For an equilateral triangle, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pedal Triangle
In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the three sides of the triangle (these may need to be produced, i.e., extended). Label ''L'', ''M'', ''N'' the intersections of the lines from ''P'' with the sides ''BC'', ''AC'', ''AB''. The pedal triangle is then ''LMN''. If ABC is not an obtuse triangle, P is the orthocenter then the angles of LMN are 180°−2A, 180°−2B and 180°−2C. The location of the chosen point ''P'' relative to the chosen triangle ''ABC'' gives rise to some special cases: * If ''P = ''orthocenter, then ''LMN = '' orthic triangle. * If ''P = ''incenter, then ''LMN = ''intouch triangle. * If ''P = ''circumcenter, then ''LMN = ''medial triangle. If ''P'' is on the circumcircle of the triangle, ''LMN'' collapses to a line. This is then called the pedal line, or so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yff Center Of Congruence
In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and Peter Yff initiated the study of this triangle center in 1987. Isoscelizer An isoscelizer of an angle ''A'' in a triangle ''ABC'' is a line through points ''P''1 and ''Q''1, where ''P''1 lies on ''AB'' and ''Q''1 on ''AC'', such that the triangle ''AP''1''Q''1 is an isosceles triangle. An isoscelizer of angle ''A'' is a line perpendicular to the bisector of angle ''A''. Isoscelizers were invented by Peter Yff in 1963. Yff central triangle Let ''ABC'' be any triangle. Let ''P''1''Q''1 be an isoscelizer of angle ''A'', ''P''2''Q''2 be an isoscelizer of angle ''B'', and ''P''3''Q''3 be an isoscelizer of angle ''C''. Let ''A'B'C' '' be the triangle formed by the three isoscelizers. The four triangles ''A'P2Q3'', ''Q1B'P3'', ''P1Q2C, and ''A'B'C' '' are always similar. There is a unique set of three isoscelizers ''P''1''Q''1, ''P''2''Q''2, ''P''3''Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
