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Jacobi Point
In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle ''ABC'' and a triple of angles ''α'', ''β'', and ''γ''. This information is sufficient to determine three points ''X'', ''Y'', and ''Z'' such that ∠''ZAB'' = ∠''YAC'' = ''α'', ∠''XBC'' = ∠''ZBA'' = ''β'', and ∠''YCA'' = ∠''XCB'' = ''γ''. Then, by a theorem of , the lines ''AX, BY,'' and ''CZ'' are concurrent,Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", ''Forum Geometricorum 15'', 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdfGlenn T. Vickers, "The 19 Congruent Jacobi Triangles", ''Forum Geometricorum'' 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf at a point ''N'' called the Jacobi point. The Jacobi point is a generalization of the Fermat point, which is obtained by letting ''α'' = ''β'' = ''γ'' = 60° and triangle ''ABC'' having no angle being greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called '' Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plane by a ... [...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] |
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] |
Forum Geometricorum
''Forum Geometricorum: A Journal on Classical Euclidean Geometry'' is a peer-reviewed open-access academic journal that specializes in mathematical research papers on Euclidean geometry. It was founded in 2001, is published by Florida Atlantic University Florida Atlantic University (Florida Atlantic or FAU) is a public research university with its main campus in Boca Raton, Florida, and satellite campuses in Dania Beach, Davie, Fort Lauderdale, Jupiter, and Fort Pierce. FAU belongs to the 12-ca ..., and is indexed among others by Mathematical Reviews and . Its founding editor-in-chief was Paul Yiu, a professor of mathematics at Florida Atlantic, now retired. All papers are available online immediately upon acceptance through the journal's web site. , Forum Geometricorum is no longer accepting submissions. Prior issues are still available. See also * International Journal of Geometry References External links * {{official website, http://forumgeom.fau.edu/ Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Point
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it. The Fermat point gives a solution to the geometric median and Steiner tree problems for three points. Construction The Fermat point of a triangle with largest angle at most 120° is simply its first isogonic center or X(13), which is constructed as follows: # Construct an equilateral triangle on each of two arbitrarily chosen sides of the given triangle. # Draw a line from each new vertex to the opposite vertex of the original triangle. # The two lines intersect at the Fermat point. An alternative method is the following: # On each of two arbitrarily chosen sides, construct an isosceles triangle, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangular Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Areal Coordinates
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or ''barycenter'') of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex. Every point has barycentric coordinates, and their sum is not zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity. Barycentric coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kiepert's Hyperbola
In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers. The name "Napoleon points" has also been applied to a different pair of triangle centers, better known as the isodynamic points. Definition of the points First Napoleon point Let ''ABC'' be any given plane triangle. On the sides ''BC'', ''CA'', ''AB'' of the triangle, construct outwardly drawn equilateral triangles ''DBC'', ''ECA'' and ''FAB'' respectively. Let the centroids of these triangles be ''X'', ''Y'' and ''Z'' respectively. Then the lines ''AX'', ''BY'' and ''CZ'' are concurrent. The point of concurrence ''N1'' is the first Napoleon point, or the outer Nap ... [...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] |
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