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Brahmagupta Formula
In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides; its generalized version (Bretschneider's formula) can be used with non-cyclic quadrilateral. Heron's formula can be thought as a sub-case of the Brahmagupta's formula. Formula Brahmagupta's formula gives the area of a cyclic quadrilateral whose sides have lengths , , , as : K=\sqrt where , the semiperimeter, is defined to be : s=\frac. This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula. If the semiperimeter is not used, Brahmagupta's formula is : K=\frac\sqrt. Another equivalent version is : K=\frac\cdot Proof Trigonometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states :c^2 = a^2 + b^2 - 2ab\cos\gamma, where denotes the angle contained between sides of lengths and and opposite the side of length . For the same figure, the other two relations are analogous: :a^2=b^2+c^2-2bc\cos\alpha, :b^2=a^2+c^2-2ac\cos\beta. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 degrees, or radians), then , and thus the law of cosines reduces to the Pythagorean theorem: :c^2 = a^2 + b^2. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known. History Though the notion of the cosine was not yet developed in his time, Euclid's '' Elements'', dating back to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sam Vandervelde
Samuel Kendrick Vandervelde (born 12 February 1971) is a mathematician who, along with Sandor Lehoczky and Richard Rusczyk, is most notable for creating the Mandelbrot Competition, and being listed first under "Thanks" in the mathematical textbook '' The Art of Problem Solving''. Contributions to mathematics Vandervelde contributes problems to the USA Math Olympiad. He was a member of the 1989 United States International Mathematical Olympiad team. He was a grader at the Mathematical Olympiad Program, an intensive summer camp that prepares top high school students for the International Math Olympiad. Vandervelde founded the Stanford Math Circle. He was on the math faculty at St. Lawrence University from 2007 to 2015, and is currently the Head of School and math teacher at Proof School, a private day school in San Francisco for kids who love math. He is also a regular instructor at and board member of MathPath, and has published two books, ''Bridge to Higher Mathematics'' and ''Circ ... [...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] |
Ptolemy's Theorem
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. If the vertices of the cyclic quadrilateral are ''A'', ''B'', ''C'', and ''D'' in order, then the theorem states that: : , \overline, \cdot , \overline, =, \overline, \cdot , \overline, +, \overline, \cdot , \overline, where the vertical lines denote the lengths of the line segments between the named vertices. This relation may be verbally expressed as follows: :''If a quadrilateral is inscribable in a circle then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides.'' Moreover, the converse of Ptolemy's theorem i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julian Coolidge
Julian Lowell Coolidge (September 28, 1873 – March 5, 1954) was an American mathematician, historian and a professor and chairman of the Harvard University Mathematics Department. Biography Born in Brookline, Massachusetts, he graduated from Harvard University and Oxford University. Between 1897 and 1899 Julian Coolidge taught at the Groton School where one of his students was Franklin D. Roosevelt. He left the private school to accept a teaching position at Harvard and in 1902 was given an assistant professorship, but took two years off to further his education with studies in Turin, Italy before receiving his doctorate from the University of Bonn. Julian Coolidge then returned to teach at Harvard where he remained for his entire academic career, interrupted only by a year at the Sorbonne in Paris as an exchange professor. During World War I, he served with the U.S. Army's Overseas Expeditionary Force in France, rising to the rank of major. In 1919, he was awarded a Knight of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inscribed Angle
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc. The inscribed angle theorem appears as Proposition 20 on Book 3 of Euclid's ''Elements''. Theorem Statement The inscribed angle theorem states that an angle ''θ'' inscribed in a circle is half of the central angle 2''θ'' that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle. Proof Inscribed angles where one chord is a diameter Let ''O'' be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them ''V'' an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supplementary Angles
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection. ''Angle'' is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. History and etymology The word ''angle'' comes from the Latin word ''angulus'', meaning "corner"; cognate words are the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Identity
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. Pythagorean identities The basic relationship between the sine and cosine is given by the Pythagorean identity: :\sin^2\theta + \cos^2\theta = 1, where \sin^2 \theta means (\sin \theta)^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the '' Khaṇḍakhādyaka'' ("edible bite", dated 665), a more practical text. Brahmagupta was the first to give rules for computing with '' zero''. The texts composed by Brahmagupta were in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived. In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. Life and career Brahmagupta was born in 598 CE according to his own statement. He lived in ''Bhillamāla'' in Gurjaradesa (modern Bhinmal in Rajasthan, India) during the reign of the Chavda dynasty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three edges and three 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- collinear, determine a unique triangle and simultaneously, a unique 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 Euclid's Elements. The names used for modern classification are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
