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Topic: Brahmagupta theorem

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	Brahmagupta - Wikipedia, the free encyclopedia
		Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer.
		Brahmagupta attempted at constructing a square of area equalling that of a circle by assuming that pi would converge at sqrt(10).
		Some of the important contributions made by Brahmagupta in astronomy are: methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon.
	en.wikipedia.org /wiki/Brahmagupta (504 words)

	Mathwords Page 12
		The theorem states that the sum of the distances from the circumcenter, O, to the three sides is equal to the sum of the radii of the incircle and the circumcircle.
		The theorem states that in an equilateral triangle, the sum of the perpendicular distances to the sides is equal to the altitude of the triangle.
		The theorem can be generalized to a regular n-gon to state, for any point P interior to a regular n-gon, the sum of the perpendicular distances to the n sides is n times the apothem of the figure.
	www.pballew.net /arithm12.html (3982 words)

	8 III. Brahmagupta, and the influence on Arabia
		Brahmagupta was born in 598 AD, possibly in Ujjain (possibly a native of Sind) and was the most influential and celebrated mathematician of the Ujjain school.
		Ptolemy 'pre-dated' Brahmagupta by 500 years, so it is wholly reasonable to attribute the 'discovery' of these rules to him.
		Without a doubt, Brahmagupta made remarkable contributions to mathematics (and astronomy) and his work continued to be influential for many centuries.
	www-history.mcs.st-and.ac.uk /history/Projects/Pearce/Chapters/Ch8_3.html (1148 words)

	Brahmagupta's Theorem
		Orthogonality plays an important role in both the formulation and the proof of the theorem.
		It's therefore a curiosity that the theorem admits a generalization that does not require the diagonals to be orthogonal.
		In the more general case the four lines are still concurrent, but they no longer meet at the intersection of the diagonals.
	www.cut-the-knot.org /Curriculum/Geometry/Brahmagupta.shtml (292 words)

	Brahmagupta's Formula for the Area of a Cyclic Quadrilateral
		Brahmagupta's formula is provides the area A of a cyclic quadrilateral (i.e., a simple quadrilateral that is inscribed in a circle) with sides of length a, b, c, and d as
		Consider Brahmagupta's formula as one side, say the one of length d wnlog, varies and approaches zero in length.
		Use Brahmagupta's formula to develop equations for the length of the two diagonals of the quadrilateral.
	jwilson.coe.uga.edu /emt725/brahmagupta/brahmagupta.html (414 words)

	Indian mathematics - Wikipedia, the free encyclopedia
		According to historian Albert Burk, this is the original proof of the theorem, and Pythagoras copied it on his visit to India.
		Stated Rolle's theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis).
		An outstanding version of the mean value theorem, which is the most important result in differential calculus and one of the most important theorems in mathematical analysis.
	www.higiena-system.com /wiki/link-Indian_Mathematics (5713 words)

	PlanetMath: cyclic quadrilateral
		A necessary and sufficient condition for a quadrilateral to be cyclic, is that the sum of a pair of opposite angles be equal to
		One of the main results about these quadrilaterals is Ptolemy's theorem.
		Cross-references: Brahmagupta's formula, sides, Ptolemy's theorem, angles, opposite, sum, circle, lie on, vertices, quadrilateral
	planetmath.org /encyclopedia/CyclicQuadrilateral.html (166 words)

	Chinese Remainder Theorem
		Chinese Remainder Theorem (CRT) The following problem was posed by Sunzi [Sun Tsu] (4th century AD) in the book Sunzi Suanjing: There are certain things whose number is unknown.
		Oystein Ore mentions another puzzle with a dramatic element from Brahma-Sphuta-Siddhanta (Brahma's Correct System) by Brahmagupta (born 598 AD): An old woman goes to market and a horse steps on her basket and crashes the eggs.
		Theorem Two simultaneous congruences n = n1 (mod m1) and n = n2 (mod m2) are only solvable when n1 = n2 (mod gcd(m1,m2)).
	www.chinapage.com /math/crt.html (863 words)

	hom
		During this period we begin to see the emergence of difference equations, the following is a history of those equations and the mathematicians that influenced them.
		The earliest major Indian mathematician was known as Brahmagupta.
		Brahmagupta was also the first person to publish a sinus table for any angle.
	www.math.uri.edu /~kulenm/diffeqaturi/m381f00fp/karen/karenmp.htm (656 words)

	Euler's Inequality
		One of the oldest inequalities about triangles is that relating the radii of the circumcircle and incircle.
		Theorem 1 (Euler 1765) Let O and I be the circumcenter and incenter, respectively, of a triangle with circumradius R and inradius r; let d be the distance OI.
		Theorem 2 In a triangle with circumradius R and inradius r, R
	math.berkeley.edu /~stankova/MathCircle/Joyce/trig/node1.html (298 words)

	The Complete Quadrilateral (Site not responding. Last check: 2007-09-09)
		The mathematical object that goes by the name of complete quadrilateral is neither complete nor quadrilateral, at least not in the sense in which the word "quadrilateral" appears in, say, Brahmagupta's theorem about cyclic quadrilaterals.
		First of all, we have the Theorem of Complete Quadrilateral: the midpoints of the three diagonals are collinear.
		Next we consider the four triangles formed by the four lines (omitting one of them at a time.) The orthocenters of the triangles are collinear and the line (Ortholine in the applet) is perpendicular to the line (Midline in the applet) of the three mid-diagonals.
	manifesto.cut-the-knot.org /ctk/CompleteQuadrilateral.shtml (1612 words)

	Pythagoras Triplets :: curiousmath :: math is an attitude
		However, the geometric theorems he and his followers developed had certainly made a big impact on modern geometry.
		His most well-known theorem in geometry, the Pythagoras Theorem, states that, for a right-angled triangle represented by three sides, a, b and c, where a and b form the right angle, and c is the hypotenuse, the equation:
		However, the greatest formula devised by Brahmagupta in the year 628, according to Heinz Becker Neumuenster, provides ALL the triplets involving a particular number greater than 2, whether it is odd or even.
	www.curiousmath.com /modules.php?op=modload&name=News&file=article&sid=45&POSTNUKESID=544f6da708946f31bf22fb932ff9094c (754 words)

	Elementary Geometry for College Students, 3e
		Heron's Theorem can be treated as a corollary of another theorem, Brahmagupta's Theorem, which can be used to calculate the area of a cyclic quadrilateral.
		A cyclic quadrilateral is one that is inscribed in a circle; as we saw in Chapter 7, Section 3, not all quadrilaterals are cyclic.
		We have provided his theorem and accompanying information to give you a better understanding of his work.
	college.hmco.com /mathematics/alexander/elementary_geometry/3e/instructors/brahma.html (141 words)

	A Math Forum Webmaster Exchange
		At first, I felt the disappointment that Grace must have felt when she looked for WWW resources on this theorem and found nothing helpful.
		With respect to Grace, who wrote in with the question, her identifying the corollary of the theorem she sought to prove was the difference between me shrugging my shoulders and me really answering her question by pursuing it from another angle.
		After corresponding with Grace, I realized a need for a reference on the details of the theorem that she set out to prove, since I did not find such a Web resource immediately.
	mathforum.org /help/webmaster/bramputa.html (2011 words)

	The Session: Shop - Product info
		But the topic follows, with clear demarcation, that of basics of composition and hence can be omitted cleanly.
		Elegant results, some of which I had not been familiar with, such as Brahmagupta's theorem, are developed in some exercises.
		In the section on exponential growth and decay, the logistic growth model is developed in the exercises.
	www.thesession.org /shop/display.php/0534352758 (559 words)

	[No title]
		Use the Internet to find: Hero’s Theorem for the area of a triangle.
		Brahmagupta’s Formula for the area of a cyclic quadrilateral.
		Extend the formulas above to produce experimental formulas to find the areas of pentagons and hexagons.
	www.amphi.com /~technology/standards/lessons/faulkner4ho1.doc (100 words)

	I am looking for a theorum - od[forum]
		A theorem to work out the largest quadrilateral that can be made out of any given 4 sides
		But I can’t make enough sense out of it as to how to turn 4 side lengths into the quadrilateral that is the largest that can be formed from the given sides with all the angles.
		Check Brahmagupta's formula on mathworld, which allows to calulate the area of any given quadrilateral:
	www.odforce.net /forum/index.php?showtopic=2216 (1249 words)

	TLCF Lesson Plan
		Students will investigate whether they can expand Hero’s Theorem and Brahmagupta’s Formula to find the area of any convex pentagon/hexagon.
		Compare the areas of pentagons/hexagons using the original formulas to the areas of the same pentagon/hexagon using the extended formulas.
		Use the appropriate technology to prepare a professional document to solve whether the theorems can be extended to include pentagons and hexagons.
	www.amphi.com /~technology/standards/lessons/faulkner4.html (353 words)

	Four Concurrent Lines in a Cyclic Quadrilateral
		Point T of concurrency is known as the anticenter of the quadrilateral ABCD.
		(This looks very much as a generalization of Brahmagupta's theorem.
		Now, on one hand, the center of gravity K lies on PR by Varignon's theorem.
	www.cut-the-knot.org /Curriculum/Geometry/Brahmagupta2.shtml (395 words)

	A Proof of the Pythagorean Theorem From Heron's Formula (Site not responding. Last check: 2007-09-09)
		Let the sides of a triangle have lengths a,b and c.
		For a quadrilateral with sides a, b, c and d inscribed in a circle there exists a generalization of Heron's formula discovered by Brahmagupta.
		Since any triangle is inscribable in a circle, we may let one side, say d, shrink to 0.
	manifesto.cut-the-knot.org /pythagoras/herons.shtml (230 words)

	Timeline of mathematics
		600s - Brahmagupta invents the method of solving indeterminate equations of the second degree and is the first to use algebra to solve astronomical problems.
		628 - Brahmagupta writes the Brahma-sphuta-siddhanta, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed.
		1100s - Bhaskara Acharya conceives differential calculus, and also develops Rolle's theorem, Pell's equation, a proof for the Pythagorean Theorem, proves that division by zero is infinity, computes π to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places
	www.danceage.com /biography/sdmc_Timeline_of_mathematics (4427 words)

	Dun dun da duh... HARD MATH PROBLEM!
		I call it the "Generalized Brahmagupta's Theorem," though it's really nothing.
		Bretschneider's Theorem finds the area of a quadrilateral from the diagonals and the sides whereas the GBT does it from the sides and a pair of opposite angles.
		The proof to these theorems are just a bunch of law of cosines and long algebraic manipulations...
	www.collegeconfidential.com /discus/messages/69/26543.html (722 words)

	Ethnomathematics Digital Library (EDL) (Site not responding. Last check: 2007-09-09)
		A new version of Miquel’s Pentagram Theorem, first published by Auguste Miquel in 1838, is now available, made into an interactive proof with animation and key theorems.
		Other theorems, problems, and proofs, as well as a section devoted to Inca Geometry can be explored.
		This article explores the potential for improving the quality of mathematics education by using a widespread decorative motif relevant to students' cultural environment.
	www.ethnomath.org /search/browseResources.asp?type=subject&id=336 (623 words)

	From Gutenberg to the Internet: Timeline 600 to 699
		The Library of Maktabat al-Jami` al-Kabir (Maktabat al-Awqaf), The Great Mosque, San`a', Yemen, built in the sixth year of Muhammad's Hijra, contains about 40 Qu'rans dating from the first century of hijra.
		Brahmagupta writes Brahmasphutasiddhanta (The Opening of the Universe).
		By this time a base 10 numeral system with nine symbols is widely used in India, and the concept of zero (represented by a dot) is known.
	www.historyofmedicine.com /G2I/docs/timeline/timeline_600_699.shtml (795 words)

	Table of contents for Library of Congress control number 2001042958
		Bibliographic record and links to related information available from the Library of Congress catalog.
		1 The Theorem of Pythagoras 1 1.1 Arithmetic and Geometry..........
		110 7.5 Construction of Equations and B6zout's Theorem.
	www.loc.gov /catdir/toc/fy031/2001042958.html (390 words)

	Hints to Some Exercises
		The general solution is found by adding appropriate multiples of a parameter k (see Brahmagupta's Theorem, p.
		(which includes the parameter k) and so the general solution is obtained (again by Brahmagupta's Theorem) by adding appropriate multiples of a new parameter l to the values of y and z.
		The third requires that you use Proth's Theorem (p.
	www.cs.xu.edu /math/math302/04f/hints.html (2622 words)

	EMAT 8990
		Click here for a GSP model to investigate.
		Brahmagupta's Theorem: In a cyclic quadrilateral having perpendicular diagonals, the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.
		Note: Brahmagupta was a Hindu mathematician 628 A.D. The position of the anticenter, the point of intersection of the diagonals of the cyclic quadrilateral, in the particular case of a cyclic quadrilateral with perpendicular diagonals was the discovery of Brahmagupta.
	jwilson.coe.uga.edu /EMT668/EMAT6680.2000/Westmoreland/gems/cyclicquads/cyclicquads.html (286 words)

	Interactive Mathematics Miscellany and Puzzles, Geometry (Site not responding. Last check: 2007-09-09)
		Golomb's inductive proof of a tromino theorem [Java]
		Menelaus Theorem: Proofs Ugly and Elegant - A. Einstein's View [Java]
		Pascal Lines: Steiner and Kirkman Theorems II [Java]
	manifesto.cut-the-knot.org /geometry.shtml (588 words)

	Wild Egg Books
		Using rational trigonometry as a basis, a new form of metrical Euclidean geometry is constructed, incorporating most of high school geometry, but extending in exciting new directions.
		This new Universal Geometry contains similiar triangles, Heron's formula, centroids, circumcenters, orthocenters, cyclic quadrilaterals and Brahmagupta's theorem, circles, parabolas, the Euler line, tangent lines and tangent conics, nine point circles, and much more---all over a general field!
		Examples are taken not only from the usual decimal numbers, but also finite fields and the complex numbers.
	wildegg.com /products.htm (808 words)

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