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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^ ... [...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 \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mass Point Geometry
Mass point geometry, colloquially known as mass points, is a problem-solving technique in geometry which applies the physical principle of the center of mass to geometry problems involving triangles and intersecting cevians. All problems that can be solved using mass point geometry can also be solved using either similar triangles, vectors, or area ratios, but many students prefer to use mass points. Though modern mass point geometry was developed in the 1960s by New York high school students, the concept has been found to have been used as early as 1827 by August Ferdinand Möbius in his theory of homogeneous coordinates. Definitions The theory of mass points is defined according to the following definitions:H. S. M. Coxeter, ''Introduction to Geometry'', pp. 216-221, John Wiley & Sons, Inc. 1969 * Mass Point - A mass point is a pair (m, P), also written as mP, including a mass, m, and an ordinary point, P on a plane. * Coincidence - We say that two points mP and nQ coincid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stewart's Theorem
In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746. Statement Let be the lengths of the sides of a triangle. Let be the length of a cevian to the side of length . If the cevian divides the side of length into two segments of length and , with adjacent to and adjacent to , then Stewart's theorem states that :b^2m + c^2n = a(d^2 + mn). A common mnemonic used by students to memorize this equation (after rearranging the terms) is: :\underset = \!\!\!\!\!\! \underset The theorem may be written more symmetrically using signed lengths of segments. That is, take the length to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. In this formulation, the theorem states that if are collinear points, and is any point, then :\left(\overline^2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh's Theorem
In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and AB, then writing \tfrac = x, \tfrac = y, and \tfrac = z, the signed area of the triangle formed by the cevians AD, BE, and CF is : S_ \frac, where S_ is the area of the triangle ABC. This theorem was given by Edward John Routh on page 82 of his ''Treatise on Analytical Statics with Numerous Examples'' in 1896. The particular case x = y = z = 2 has become popularized as the one-seventh area triangle. The x = y = z = 1 case implies that the three medians are concurrent (through the centroid). Proof Suppose that the area of triangle ABC is 1. For triangle ABD and line FRC using Menelaus's theorem, We could obtain: :\frac \times \frac \times \frac = 1 Then \frac = \frac \times \frac = \frac So the area of triangle ARC is: :S_ = \frac S_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple ... [...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 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Altitude (triangle)
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ''extended base'' of the altitude. The intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel/circle (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter; if the length of the string was exact, it would equal the perimeter. Formulas The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with \int_0^L \mathrms, where L is the length of the path and ds is an infinitesimal line element. Both of these must be replaced by algebraic forms in order to be practically calculated. If the perimeter is given as a closed piecewise smooth plane curv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitter (geometry)
In Euclidean geometry, a splitter is a line segment through one of the vertices of a triangle (that is, a cevian) that bisects the perimeter of the triangle. They are not to be confused with cleavers, which also bisect the perimeter but instead emanate from the midpoint of one of the triangle's sides. Properties The opposite endpoint of a splitter to the chosen triangle vertex lies at the point on the triangle's side where one of the excircles of the triangle is tangent to that side. This point is also called a splitting point of the triangle. It is additionally a vertex of the extouch triangle and one of the points where the Mandart inellipse is tangent to the triangle side. The three splitters concur at the Nagel point of the triangle, which is also called its splitting center. Generalization Some authors have used the term "splitter" in a more general sense, for any line segment that bisects the perimeter of the triangle. Other line segments of this type include the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morley Triangle
In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles. Proofs There are many proofs of Morley's theorem, some of which are very technical. Several early proofs were based on delicate trigonometric calculations. Recent proofs include an algebraic proof by extending the theorem to general fields other than characteristic three, and John Conway's elementary geometry proof. The latter starts with an equilateral triangle and shows that a triangle may be built around it which will be similar to any selected triangle. Morley's theorem does not hold in spherical and hyperbolic geometry. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
