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Concurrent Lines
In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point. The set of all lines through a point is called a ''pencil'', and their common intersection is called the '' vertex'' of the pencil. In any affine space (including a Euclidean space) the set of lines parallel to a given line (sharing the same direction) is also called a ''pencil'', and the vertex of each pencil of parallel lines is a distinct point at infinity; including these points results in a projective space in which every pair of lines has an intersection. 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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisection
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, ^ ... [...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 w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incircle
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spieker Circle
In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the center of mass of the uniform-density boundary of triangle. The Spieker center is also the point where all three cleavers of the triangle (perimeter bisectors with an endpoint at a side's midpoint) intersect each other. History The Spieker circle and Spieker center are named after Theodor Spieker, a mathematician and professor from Potsdam, Germany. In 1862, he published , dealing with planar geometry. Due to this publication, influential in the lives of many famous scientists and mathematicians including Albert Einstein, Spieker became the mathematician for whom the Spieker circle and center were named. Construction To find the Spieker circle of a triangle, the medial triangle must first be constructed from the midpoints of each side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cleaver (geometry)
In geometry, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. They are not to be confused with '' splitters'', which also bisect the perimeter, but with an endpoint on one of the triangle's vertices instead of its sides. Construction Each cleaver through the midpoint of one of the sides of a triangle is parallel to the angle bisectors at the opposite vertex of the triangle. The broken chord theorem of Archimedes provides another construction of the cleaver. Suppose the triangle to be bisected is , and that one endpoint of the cleaver is the midpoint of side . Form the circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ... of and let be the midpoint of the arc of the cir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Gazette
''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journal was established in 1894 by Edward Mann Langley as the successor to the ''Reports of the Association for the Improvement of Geometrical Teaching''. William John Greenstreet was its editor-in-chief for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors-in-chief The following persons are or have been editor-in-chief: Abstracting and indexing The journal is abstracted and indexed in EBSCO databases, Emerging Sources Citation Index, Scopus Scopus is a scientific abstract and citation database, launched by the academic publisher Elsevier as a competitor to older Web of Science in 2004. The ensuing competition between the two databases has been characterized as "intense" and is c ..., and zbMA ... [...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] |
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] |
Incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle. Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one of the four that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers.. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides. Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle. The term ''angle'' is also used for the size, magnitude (mathematics), magnitude or Physical quantity, quantity of these types of geometric figures and in this context an a ... [...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] |
