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Exeter Point
In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986. This is one of the recent triangle centers, unlike the classical triangle centers like centroid, incenter, and Steiner point. Definition Exeter point is defined as follows. :Let ''ABC'' be any given triangle. Let the medians through the vertices ''A'', ''B'', ''C'' meet the circumcircle of triangle ''ABC'' at ''A' '', ''B' '' and ''C' '' respectively. Let ''DEF'' be the triangle formed by the tangents at ''A'', ''B'', and ''C'' to the circumcircle of triangle ''ABC''. (Let ''D'' be the vertex opposite to the side formed by the tangent at the vertex ''A'', ''E'' be the vertex opposite to the side formed by the tangent at the vertex ''B'', and ''F'' be the vertex opposite t ... [...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] |
Median (geometry)
In geometry, a median of a triangle is a line segment joining a vertex (geometry), vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's triangle centroid, centroid. In the case of isosceles and equilateral triangles, a median angle bisector, bisects any angle at a vertex whose two adjacent sides are equal in length. The concept of a median extends to tetrahedron, tetrahedra. Relation to center of mass Each median of a triangle passes through the triangle's centroid, which is the center of mass of an infinitely thin object of uniform density coinciding with the triangle. Thus the object would balance on the intersection point of the medians. The centroid is twice as close along any median to the side that the median intersects as it is to the vertex it emanates from. Equal-area division Each median divides the area of the triangle in half; hence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Centre
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Longchamps Point
In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.. Definition Let the given triangle have vertices A, B, and C, opposite the respective sides a, b, and c, as is the standard notation in triangle geometry. In the 1886 paper in which he introduced this point, de Longchamps initially defined it as the center of a circle \Delta orthogonal to the three circles \Delta_a, \Delta_b, and \Delta_c, where \Delta_a is centered at A with radius a and the other two circles are defined symmetrically. De Longchamps then also showed that the same point, now known as the de Longchamps point, may be equivalently defined as the orthocenter of the anticomplementary triangle of ABC, and that it is the reflection of the orthocenter of ABC around the circumcenter. The Steiner circle of a triangle is concentric with the nine-point ... [...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] |
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] |
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] |
Concurrent Lines
In geometry, lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. They are in contrast to parallel lines. Examples Triangles In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors: * A triangle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter. * Angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter. * Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid. * Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter. Other sets of lines associated with a triangle are concurrent as well. For example: * Any median ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcircle
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 sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exeter Point
In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986. This is one of the recent triangle centers, unlike the classical triangle centers like centroid, incenter, and Steiner point. Definition Exeter point is defined as follows. :Let ''ABC'' be any given triangle. Let the medians through the vertices ''A'', ''B'', ''C'' meet the circumcircle of triangle ''ABC'' at ''A' '', ''B' '' and ''C' '' respectively. Let ''DEF'' be the triangle formed by the tangents at ''A'', ''B'', and ''C'' to the circumcircle of triangle ''ABC''. (Let ''D'' be the vertex opposite to the side formed by the tangent at the vertex ''A'', ''E'' be the vertex opposite to the side formed by the tangent at the vertex ''B'', and ''F'' be the vertex opposite t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane (geometry)
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane. Euclidean geometry Euclid set forth the first great landmark of mathematical thought, an axiomatic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner Point (triangle)
In triangle geometry, the Steiner point is a particular point associated with a triangle. It is a triangle center and it is designated as the center X(99) in Clark Kimberling's Encyclopedia of Triangle Centers. Jakob Steiner (1796–1863), Swiss mathematician, described this point in 1826. The point was given Steiner's name by Joseph Neuberg in 1886. Definition The Steiner point is defined as follows. (This is not the way in which Steiner defined it.) :Let be any given triangle. Let be the circumcenter and be the symmedian point of triangle . The circle with as diameter is the Brocard circle of triangle . The line through perpendicular to the line intersects the Brocard circle at another point . The line through perpendicular to the line intersects the Brocard circle at another point . The line through perpendicular to the line intersects the Brocard circle at another point . (The triangle is the Brocard triangle of triangle .) Let be the line through parallel t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
