|
List Of Triangle Topics
This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrix, triangular matrices, or concretely in physical space. It does not include metaphors like love triangle in which the word has no reference to the geometric shape. Geometry *Triangle Talk:Triangle, *Acute and obtuse triangles Talk:Acute and obtuse triangles, *Altern base Talk:Altern base, *Altitude (triangle) Talk:Altitude (triangle), *Bisection#Area bisectors and perimeter bisectors, Area bisector of a triangle *Bisection#Angle bisector, Angle bisector of a triangle *Angle bisector theorem Talk: Angle bisector theorem, *Apollonius point Talk:Apollonius point, *Apollonius' theorem Talk:Apollonius' theorem, *Automedian triangle Talk:Automedian triangle, *Barrow's inequality *Barycentric coordinates (mathematics) *Bernoulli's quadrisection problem Talk:Bernoulli's ... [...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] |
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 equila ... [...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 of and let be the midpoint of the arc of the circumcircle from to through . Then the other endpoint of the cleaver is the closest point of the triangle to , and can be found by dropping a perpendicular from to the longer of the two sides and . Related figures The three cleavers concur at a point, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clawson Point
The Clawson point is a special point in a planar triangle defined by the trilinear coordinates \tan(\alpha):\tan(\beta):\tan(\gamma) ( Kimberling number X(19)), where \alpha, \beta, \gamma are the interior angles at the triangle vertices A, B, C. It is named after John Wentworth Clawson, who published it 1925 in the American Mathematical Monthly. Geometrical constructions There are at least two ways to construct the Clawson point, which also could be used as coordinate free definitions of the point. In both cases you have two triangles, where the three lines connecting their according vertices meet in a common point, which is the Clawson point. Construction 1 For a given triangle \triangle ABC let \triangle H_aH_bH_c be its orthic triangle and \triangle T_aT_bT_c the triangle formed by the outer tangents to its three excircles. These two triangles are similar and the Clawson point is their center of similarity, therefore the three lines T_aH_a, T_bH_b, T_cH_c connecting thei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumscribed Circle
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] |
Circumconic And Inconic
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html Suppose are distinct non-collinear points, and let denote the triangle whose vertices are . Following common practice, denotes not only the vertex but also the angle at vertex , and similarly for and as angles in . Let a= , BC, , b=, CA, , c=, AB, , the sidelengths of . In trilinear coordinates, the general circumconic is the locus of a variable point X = x:y:z satisfying an equation :uyz + vzx + wxy = 0, for some point . The isogonal conjugate of each point on the circumconic, other than , is a point on the line :ux + vy + wz = 0. This line meets the circumcircle of in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cevian
In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name. Length Stewart's theorem The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length is given by the formula :\,b^2m + c^2n = a(d^2 + mn). Less commonly, this is also represented (with some rearrangement) by the following mnemonic: :\underset = \!\!\!\!\!\! \underset Median If the cevian happens to be a median (thus bisecting a side), its length can be determined from the formula :\,m(b^2 + c^2) = a(d^2 + m^2) or :\,2(b^2 + c^2) = 4d^2 + a^2 since :\,a = 2m. Hence in this case :d= \frac\sqrt2 . Angle bisector If the cevian happens to be an angle bisector, its length obeys the formulas :\,(b + c)^2 = a^2 \le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ceva's Theorem
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are known as cevians.) Then, using signed lengths of segments, :\frac \cdot \frac \cdot \frac = 1. In other words, the length is taken to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. For example, is defined as having positive value when is between and and negative otherwise. Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field. A slightly adapted converse is also true: If points are chosen on respectively so that : \frac \cd ... [...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 surface of 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" is of recent coinage (1814). It is used as a substitute for the older te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalogue Of Triangle Cubics
The Catalogue of Triangle Cubics is an online resource containing detailed information about more than 1200 cubic curves in the plane of a reference triangle. The resource is maintained by Bernard Gilbert. Each cubic in the resource is assigned a unique identification number of the form "Knnn" where "nnn" denotes three digits. The identification number of the first entry in the catalogue is "K001" which is the Neuberg cubic of the reference triangle . The catalogue provides, among other things, the following information about each of the cubics listed: * Barycentric equation of the curve *A list of triangle centers which lie on the curve *Special points on the curve which are not triangle centers *Geometric properties of the curve *Locus properties of the curve *Other special properties of the curve *Other curves related to the cubic curve *Plenty of neat and tidy figures illustrating the various properties *References to literature on the curve The equations of some of the cubics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carnot's Theorem (perpendiculars)
Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection. The theorem can also be thought of as a generalization of the Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t .... Theorem For a triangle \triangle ABC with sides a, b, c consider three lines that are perpendicular to the triangle sides and intersect in a common point F. If P_a, P_b, P_c are the pedal points of those three perpendiculars on the sides a, b, c, then the following equation holds: : , AP_c, ^2+, BP_a, ^2+, CP_b, ^2=, BP_c, ^2+, CP_a, ^2+, AP_b, ^2 The converse of the statement above is true as well, that is if the equation holds for the peda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carnot's Theorem (inradius, Circumradius)
In Euclidean geometry, Carnot's theorem states that the sum of the signed distances from the circumcenter ''D'' to the sides of an arbitrary triangle ''ABC'' is :DF + DG + DH = R + r,\ where ''r'' is the inradius and ''R'' is the circumradius of the triangle. Here the sign of the distances is taken to be negative if and only if the open line segment ''DX'' (''X'' = ''F'', ''G'', ''H'') lies completely outside the triangle. In the diagram, ''DF'' is negative and both ''DG'' and ''DH'' are positive. The theorem is named after Lazare Carnot (1753–1823). It is used in a proof of the Japanese theorem for concyclic polygons. References *Claudi Alsina, Roger B. Nelsen: ''When Less is More: Visualizing Basic Inequalities''. MAA, 2009, , 99*Frédéric Perrier: ''Carnot's Theorem in Trigonometric Disguise''. The Mathematical Gazette, Volume 91, No. 520 (March, 2007), pp. 115–117JSTOR *David Richeson''The Japanese Theorem for Nonconvex Polygons – Carnot's Theorem'' Convergenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
