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Isotomic Conjugate
In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of . Construction We assume that is not collinear with any two vertices of . Let be the points in which the lines meet sidelines (extended if necessary). Reflecting in the midpoints of sides will give points respectively. The isotomic lines joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the ''isotomic conjugate'' of . Coordinates If the trilinears for are , then the trilinears for the isotomic conjugate of are :a^p^ : b^q^ : c^r^, where are the side lengths opposite vertices respectively. Properties The isotomic conjugate of the centroid of triangle is the centroid itself. The isotomic conjugate of the ... [...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] |
Reflection (geometry)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term ''reflection'' is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimensional space whose endpoints are A = (a_1, a_2, \dots , a_n) and B = (b_1, b_2, \dots , b_n) is given by :\frac. That is, the ''i''th coordinate of the midpoint (''i'' = 1, 2, ..., ''n'') is :\frac 2. Construction Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction. The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect). The point where the line connecting the cusps intersects the segment is then the midpoint of the segment. It is more ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Side
In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a side arises in various contexts. Triangle In an obtuse triangle, the altitudes from the acute angled vertices intersect the corresponding extended base sides but not the base sides themselves. The excircles of a triangle, as well as the triangle's inconics that are not inellipses, are externally tangent to one side and to the other two extended sides. Trilinear coordinates locate a point in the plane by its relative distances from the extended sides of a reference triangle. If the point is outside the triangle, the perpendicular from the point to the sideline may meet the sideline outside the triangle—that is, not on the actual side of the triangle. In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 20 ... [...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] |
Trilinear Coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ... [...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] |
Symmedian Point
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry".. Isogonality Many times in geometry, if we take three special lines through the vertices of a triangle, or ''cevians'', then their reflections about the corresponding angle bisectors, called ''isogonal lines'', will also have interesting properties. For instance, if three cevians of a triangle intersect ... [...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 wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gergonne Point
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 the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nagel Point
In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurrency of all three of the triangle's splitters. Construction Given a triangle , let be the extouch points in which the -excircle meets line , the -excircle meets line , and the -excircle meets line , respectively. The lines concur in the Nagel point of triangle . Another construction of the point is to start at and trace around triangle half its perimeter, and similarly for and . Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments are called the triangle's splitters. There exists an easy construction of the Nagel point. Starting from each vertex of a triangle, it suffices to carry twice the length of the opposite edge. We obtain three lines which concur at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
