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Angle Bisector Theorem
In geometry, the angle bisector theorem is concerned with the relative lengths of the two Line segment, segments that a triangle's side is divided into by a Line (geometry), line that Bisection, bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle. Theorem Consider a triangle . Let the Bisection#Angle bisector, angle bisector of angle Line-line intersection, intersect side at a point between and . The angle bisector theorem states that the ratio of the length of the line segment to the length of segment is equal to the ratio of the length of side to the length of side : :=, and conversely, if a point on the side of divides in the same ratio as the sides and , then is the angle bisector of angle . The generalized angle bisector theorem (which is not necessarily an angle bisector theorem, since the angle is not necessarily bisected into equal parts) states that if lies on the line , then :=. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle ABC With Bisector AD
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called Vertex (geometry), ''vertices'', are zero-dimensional point (geometry), points while the sides connecting them, also called Edge (geometry), ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base (geometry), ''base'', in which case the opposite vertex is called the apex (geometry), ''apex''; the shortest segment between the base and apex is the height (triangle), ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Bisector Proof
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides. Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle. The term ''angle'' is also used for the size, magnitude or quantity of these types of geometric figures and in this context an angle consists of a number and unit of measurement. Angular measure or measure of angle are sometimes used to distinguish between the measurement and f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Khan Academy
Khan Academy is an American non-profit educational organization created in 2006 by Sal Khan. Its goal is to create a set of online tools that help educate students. The organization produces short video lessons. Its website also includes supplementary practice exercises and materials for educators. It has produced over 10,000 video lessons teaching a wide spectrum of academic subjects, including mathematics, sciences, literature, history, and computer science. All resources are available free to users of the website and application. History Starting in 2004, Salman "Sal" Khan began tutoring one of his cousins in mathematics on the Internet using a service called Yahoo! Doodle Images. After a while, Khan's other cousins began to use his tutoring service. Due to the demand, Khan decided to make his videos watchable on the Internet, so he published his content on YouTube. Later, he used a drawing application called SmoothDraw, and now uses a Wacom tablet to draw using ArtRa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games. He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website ''Cut-the-Knot'' for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started ''Cut-the-Knot'' in October 1996. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circles Of Apollonius
The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection. The main uses of this term are fivefold: # Apollonius showed that a circle can be defined as the set of points in a plane that have a specified ''ratio'' of distances to two fixed points, known as foci. This Apollonian circle is the basis of the Apollonius pursuit problem. It is a particular case of the first family described in #2. # The Apollonian circles are two families of mutually orthogonal circles. The first family consists of the circles with all possible distance ratios to two fixed foci (the same circles as in #1), whereas the second family consists of all possible circles that pass through both foci. These circles form the basis o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle. Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one of the four that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers.. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the underlying principles of which he formalized. De Morgan's contributions to logic are heavily used in many branches of mathematics, including set theory and probability theory, as well as other related fields such as computer science. Biography Childhood Augustus De Morgan was born in Madurai, in the Carnatic Sultanate, Carnatic region of India, in 1806. His father was Lieutenant-Colonel John De Morgan (1772–1816), who held various appointments in the service of the East India Company, and his mother, Elizabeth (née Dodson, 1776–1856), was the granddaughter of James Dodson (mathematician), James Dodson, who computed a table of anti-logarithms (inverse logarithms). Augustus De Morgan became blind in one eye within a few months of his bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Of Alexandria
Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known about his life except for what can be found in his own writings, many of which are lost. Pappus apparently lived in Alexandria, where he worked as a Mathematics education, mathematics teacher to higher level students, one of whom was named Hermodorus.Pierre Dedron, J. Itard (1959) ''Mathematics And Mathematicians'', Vol. 1, p. 149 (trans. Judith V. Field) (Transworld Student Library, 1974) The ''Collection'', his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics that were part of the ancient mathematics curriculum, including geometry, astronomy, and mechanics. Pappus was active in a period generally considered one of stagnation in mathematical studies, where, to s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Simson
Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him.Robert Simson University of Glasgow (multi-tab page) Biography Robert Simson was born on 14 October 1687, probably the eldest of the seventeen children, all male, of John Simson, a Glasgow merchant, and Agnes, daughter of Patrick Simpson, minister of Renfrew; only six of them reached adulthood. Simson matriculated at the in 1701, intending to enter the Church. He followed the course in the faculty of arts (Latin, Greek, logic, natural phil ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus and Theaetetus (mathematician), Theaetetus, the ''Elements'' is a collection in 13 books of definitions, postulates, propositions and mathematical proofs that covers plane and solid Euclidean geometry, elementary number theory, and Commensurability (mathematics), incommensurable lines. These include Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the Compass-and-straightedge construction, construction of regular polygons and Regular polyhedra, polyhedra. Often referred to as the most successful textbook ever written, the ''Elements'' has continued to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonius's Theorem
In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side. The theorem is found as proposition VII.122 of Pappus of Alexandria's ''Collection'' (). It may have been in Apollonius of Perga's lost treatise ''Plane Loci'' (c. 200 BC), and was included in Robert Simson's 1749 reconstruction of that work. Statement and relation to other theorem In any triangle ABC, if AD is a median (, BD, = , CD, ), then , AB, ^2+, AC, ^2=2(, BD, ^2+, AD, ^2). It is a special case of Stewart's theorem. For an isosceles triangle with , AB, = , AC, , the median AD is perpendicular to BC and the theorem reduces to the Pythagorean theorem for triangle ADB (or triangle ADC). From the fact that the diagonals of a parallelogram bisect each other, the th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
