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List Of Triangle Inequalities
In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of the form "less than", "less than or equal to", "greater than", or "greater than or equal to". The parameters in a triangle inequality can be the side lengths, the semiperimeter, the angle measures, the values of trigonometric functions of those angles, the area of the triangle, the medians of the sides, the altitudes, the lengths of the internal angle bisectors from each angle to the opposite side, the perpendicular bisectors of the sides, the distance from an arbitrary point to another point, the inradius, the exradii, the circumradius, and/or other quantities. Unless otherwise specified, this article deals with triangles in the Euclidean plane. Main parameters and notation The parameters most commonly appearing in triangle inequaliti ... [...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 ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crux Mathematicorum
''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. , its editor-in-chief is Kseniya Garaschuk. The journal was established in 1975, under the name ''Eureka'', by the Carleton-Ottawa Mathematics Association, with Léo Sauvé as its first editor-in-chief. It took the name ''Crux Mathematicorum'' with its fourth volume, in 1978, to avoid confusion with another journal '' Eureka'' published by the Cambridge University Mathematical Society. The Canadian Mathematical Society took over the journal in 1985, and soon afterwards G.W. (Bill) Sands became its new editor. Bruce L. R. Shawyer took over as editor in 1996. In 1997 it merged with another journal founded in 1988, ''Mathematical Mayhem'', to become ''Crux Mathematicorum with Mathematical Mayhem''. Jim Totten became editor in 2003, and Václav (Vazz) Linek replaced him in 2008. Ross Honsberger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides ''a'', ''b'' and the hypotenuse ''c'', often called the Pythagorean equation: :a^2 + b^2 = c^2 , The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Angle
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin ''angulus rectus''; here ''rectus'' means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry. Etymology The meaning of ''right'' in ''right angle'' possibly refers to the Latin adjective ''rectus'' 'erect, straight, upright, perpendicular'. A Greek equivalent is ''orthos'' 'straight; perpendicular' (see orthogon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nesbitt's Inequality
In mathematics, Nesbitt's inequality states that for positive real numbers ''a'', ''b'' and ''c'', :\frac+\frac+\frac\geq\frac. It is an elementary special case (N = 3) of the difficult and much studied Shapiro inequality, and was published at least 50 years earlier. There is no corresponding upper bound as any of the 3 fractions in the inequality can be made arbitrarily large. Proof First proof: AM-HM inequality By the AM- HM inequality on (a+b),(b+c),(c+a), :\frac\geq\frac. Clearing denominators yields :((a+b)+(a+c)+(b+c))\left(\frac+\frac+\frac\right)\geq 9, from which we obtain :2\frac+2\frac+2\frac\geq9 by expanding the product and collecting like denominators. This then simplifies directly to the final result. Second proof: Rearrangement Suppose a \ge b \ge c , we have that :\frac 1 \ge \frac 1 \ge \frac 1 define :\vec x = (a, b, c) :\vec y = \left(\frac 1 , \frac 1 , \frac 1 \right) The scalar product of the two sequences is maximum because of the rearrangeme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit of a function is usually written as : \lim_ f(x) = L, (although a few authors may use "Lt" instead of "lim") and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or \rightarrow), as in :f(x) \to L \text x \to c, which reads "f of x tends to L as x tends to c". History Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work ''Opus Geometricum'' (1647): "The ''terminus'' of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Secrets Of Triangles
''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books. Topics The book consists of ten chapters, with the first six concentrating on triangle centers while the final four cover more diverse topics including the area of triangles, inequalities involving triangles, straightedge and compass constructions, and fractals. Beyond the classical triangle centers (the circumcenter, incenter, orthocenter, and centroid) the book covers other centers including the Brocard points, Fermat point, Gergonne point, and other geometric objects associated with triangle centers such as the Euler line, Simson line, and nine-point circle. The chapter on areas includes both trigonometric formulas and Heron's formula for computing the area of a triangle from its side lengths, and the chapter on inequalities includes the Erdős–Mordell ine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Posamentier
Alfred S. Posamentier (born October 18, 1942) is an American educator and a lead commentator on American math and science education, regularly contributing to ''The New York Times'' and other news publications. He has created original math and science curricula, emphasized the need for increased math and science funding, promulgated criteria by which to select math and science educators, advocated the importance of involving parents in K-12 math and science education, and provided myriad curricular solutions for teaching critical thinking in math. Dr. Posamentier was a member of the New York State Education Commissioner's Blue Ribbon Panel on the Math-A Regents Exams. He served on the Commissioner's Mathematics Standards Committee, which redefined the Standards for New York State. And he served on the New York City schools’ Chancellor's Math Advisory Panel. Posamentier earned a Ph.D. in mathematics education from Fordham University (1973), a master's degree in mathematics educa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of the triangle, with no side being greater than , then the triangle inequality states that :z \leq x + y , with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths ( norms): :\, \mathbf x + \mathbf y\, \leq \, \mathbf x\, + \, \mathbf y\, , where the length of the third side has been replaced by the vector sum . When and are real numbers, they can be viewed as vectors in , and the trian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. Similarly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
