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Brocard Triangle
In geometry, the Brocard triangle of a triangle is a triangle formed by the intersection of lines from a vertex to its corresponding Brocard point and a line from another vertex to its corresponding Brocard point and the other two points constructed using different combinations of vertices and Brocard points. This triangle is also called the first Brocard triangle, as further triangles can be formed by forming the Brocard triangle of the Brocard triangle and continuing this pattern. The Brocard triangle is inscribed in the Brocard circle. It is named for Henri Brocard. See also * Henri Brocard *Brocard points In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled ... Notes Triangles {{Elementary-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brocard Triangle
In geometry, the Brocard triangle of a triangle is a triangle formed by the intersection of lines from a vertex to its corresponding Brocard point and a line from another vertex to its corresponding Brocard point and the other two points constructed using different combinations of vertices and Brocard points. This triangle is also called the first Brocard triangle, as further triangles can be formed by forming the Brocard triangle of the Brocard triangle and continuing this pattern. The Brocard triangle is inscribed in the Brocard circle. It is named for Henri Brocard. See also * Henri Brocard *Brocard points In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled ... Notes Triangles {{Elementary-geometry-stub ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three edges and three 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- collinear, determine a unique triangle and simultaneously, a unique 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 Euclid's Elements. The names used for modern classification are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brocard Point
In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled ''A'', ''B'' and ''C'' in counterclockwise order, there is exactly one point ''P'' such that the line segments ''AP'', ''BP'', and ''CP'' form the same angle, ω, with the respective sides ''c'', ''a'', and ''b'', namely that : \angle PAB = \angle PBC = \angle PCA =\omega.\, Point ''P'' is called the first Brocard point of the triangle ''ABC'', and the angle ''ω'' is called the Brocard angle of the triangle. This angle has the property that :\cot\omega = \cot \alpha + \cot \beta + \cot \gamma, \, where \alpha, \, \beta, \, \gamma are the vertex angles \angle CAB, \, \angle ABC, \, \angle BCA respectively. There is also a second Brocard point, Q, in triangle ''ABC'' such that line segments ''AQ'', ''BQ'', and ''CQ'' form equal angles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brocard Circle
In geometry, the Brocard circle (or seven-point circle) is a circle derived from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this segment is a diameter). Equation In terms of the side lengths a, b, and c of the given triangle, and the areal coordinates (x,y,z) for points inside the triangle (where the x-coordinate of a point is the area of the triangle made by that point with the side of length a, etc), the Brocard circle consists of the points satisfying the equation :a^2yz+b^2zx+c^2xy=\frac\left(\frac+\frac+\frac\right). Related points The two Brocard points lie on this circle, as do the vertices of the Brocard triangle. These five points, together with the other two points on the circle (the circumcenter and symmedian), justify the name "seven-point circle". The Brocard circle is concentric with the first Lemoine circle. Special cases If the triangle is equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Brocard
Pierre René Jean Baptiste Henri Brocard (12 May 1845 – 16 January 1922) was a French meteorologist and mathematician, in particular a geometer. His best-known achievement is the invention and discovery of the properties of the Brocard points, the Brocard circle, and the Brocard triangle, all bearing his name. Contemporary mathematician Nathan Court wrote that he, along with Émile Lemoine and Joseph Neuberg, was one of the three co-founders of modern triangle geometry. He is listed as an Emeritus at the International Academy of Science, was awarded the Ordre des Palmes Académiques, and was an officer of the Légion d'honneur. He spent most of his life studying meteorology as an officer in the French Navy, but seems to have made no notable original contributions to the subject. Biography Early years Pierre René Jean Baptiste Henri Brocard was born on 12 May 1845, in Vignot (a part of Commercy), Meuse to Elizabeth Auguste Liouville and Jean Sebastien Brocard. He attended ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brocard Points
In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled ''A'', ''B'' and ''C'' in counterclockwise order, there is exactly one point ''P'' such that the line segments ''AP'', ''BP'', and ''CP'' form the same angle, ω, with the respective sides ''c'', ''a'', and ''b'', namely that : \angle PAB = \angle PBC = \angle PCA =\omega.\, Point ''P'' is called the first Brocard point of the triangle ''ABC'', and the angle ''ω'' is called the Brocard angle of the triangle. This angle has the property that :\cot\omega = \cot \alpha + \cot \beta + \cot \gamma, \, where \alpha, \, \beta, \, \gamma are the vertex angles \angle CAB, \, \angle ABC, \, \angle BCA respectively. There is also a second Brocard point, Q, in triangle ''ABC'' such that line segments ''AQ'', ''BQ'', and ''CQ'' form equal angles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
