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Congruent Isoscelizers Point
In geometry the congruent isoscelizers point is a special point associated with a plane triangle. It is a triangle center and it is listed as X(173) in Clark Kimberling's Encyclopedia of Triangle Centers. This point was introduced to the study of triangle geometry by Peter Yff in 1989. Definition 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. Let ''ABC'' be any triangle. Let ''P''1''Q''1, ''P''2''Q''2, ''P''3''Q''3 be the isoscelizers of the angles ''A'', ''B'', ''C'' respectively such that they all have the same length. Then, for a unique configuration, the three isoscelizers ''P''1''Q''1, ''P''2''Q''2, ''P''3''Q''3 are concurrent. The point of concurrence is the ''congruent isoscelizers point'' of triangle ''ABC''. Properties *The trilinear ... [...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] |
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] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...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] |
Clark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer sequences, and hymnology. Kimberling received his PhD in mathematics in 1970 from the Illinois Institute of Technology, under the supervision of Abe Sklar. Since at least 1994, he has maintained a list of triangle centers and their properties. In its current on-line form, the Encyclopedia of Triangle Centers, this list comprises tens of thousands of entries. He has contributed to ''The Hymn'', the journal of the Hymn Society in the United States and Canada; and in the '' Canterbury Dictionary of Hymnology''. Kimberling's golden triangle Robert C. Schoen has defined a "golden triangle" as a triangle with two of its sides in the golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encyclopedia Of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the list identifies 52,440 triangle centers. Each point in the list is identified by an index number of the form ''X''(''n'')—for example, ''X''(1) is the incenter. The information recorded about each point includes its trilinear and barycentric coordinates and its relation to lines joining other identified points. Links to The Geometer's Sketchpad diagrams are provided for key points. The Encyclopedia also includes a glossary of terms and definitions. Each point in the list is assigned a unique name. In cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, the 770th point in the list is named ''point Acamar''. The first 10 points listed in the Encyclopedia ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruent Isoscelizers Point
In geometry the congruent isoscelizers point is a special point associated with a plane triangle. It is a triangle center and it is listed as X(173) in Clark Kimberling's Encyclopedia of Triangle Centers. This point was introduced to the study of triangle geometry by Peter Yff in 1989. Definition 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. Let ''ABC'' be any triangle. Let ''P''1''Q''1, ''P''2''Q''2, ''P''3''Q''3 be the isoscelizers of the angles ''A'', ''B'', ''C'' respectively such that they all have the same length. Then, for a unique configuration, the three isoscelizers ''P''1''Q''1, ''P''2''Q''2, ''P''3''Q''3 are concurrent. The point of concurrence is the ''congruent isoscelizers point'' of triangle ''ABC''. Properties *The trilinear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoscelizer
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'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Construction For Congruent Isoscelizers Point
Construction is a general term meaning the art and science to form objects, systems, or organizations,"Construction" def. 1.a. 1.b. and 1.c. ''Oxford English Dictionary'' Second Edition on CD-ROM (v. 4.0) Oxford University Press 2009 and comes from Latin ''constructio'' (from ''com-'' "together" and ''struere'' "to pile up") and Old French ''construction''. To construct is the verb: the act of building, and the noun is construction: how something is built, the nature of its structure. In its most widely used context, construction covers the processes involved in delivering buildings, infrastructure, industrial facilities and associated activities through to the end of their life. It typically starts with planning, financing, and design, and continues until the asset is built and ready for use; construction also covers repairs and maintenance work, any works to expand, extend and improve the asset, and its eventual demolition, dismantling or decommissioning. The construction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
