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	Cyclic quadrilateral - Wikipedia, the free encyclopedia
		In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle.
		The area of a cyclic quadrilateral is given by Brahmagupta's formula as well as Heron's formula as long as the sides are given.
		The area of a cyclic quadrilateral is maximal among all quadrilaterals having the same side lengths.
	en.wikipedia.org /wiki/Cyclic_quadrilateral (151 words)

	cylic quadralerals
		The length of the two diagonals of a cyclic quadrilateral are related to the four sides in Ptolemy's Theorem which states (using m and n for the diagonals lengths) mn=ac+bd.
		The center of this "orthic cyclic quadrilateral" is the reflection of the circumcenter of the original quadrilateral in the anti-center.
		The anti-center of the orthic quadrilateral is the same as the anti-center of the original quadrilateral, and so the orthocenters of the triangles formed by the orthic quadrilateral are the vertices of the original cyclic quadrilateral.
	www.pballew.net /cycquad.html (1589 words)

	PlanetMath: cyclic quadrilateral
		A quadrilateral is cyclic when its four vertices lie on a circle.
		A necessary and sufficient condition for a quadrilateral to be cyclic, is that the sum of a pair of opposite angles be equal to
		This is version 4 of cyclic quadrilateral, born on 2001-10-06, modified 2002-03-09.
	planetmath.org /encyclopedia/CyclicQuadrilateral.html (166 words)

	Brahmagupta's formula - Wikipedia, the free encyclopedia
		In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle.
		In the case of non-cyclic quadrilaterals, Brahmagupta's formula can be extended by considering the measures of two opposite angles of the quadrilateral:
		It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to
	en.wikipedia.org /wiki/Brahmagupta's_formula (215 words)

	PlanetMath: cyclic ring
		A ring is a http://planetmath.org/encyclopedia/SpecialLinearGroup.html cyclic ring f its additive group is cyclic.
		This is version 15 of cyclic ring, born on 2003-03-10, modified 2003-04-02.
		There are at least two articles by the name of "cyclic", and, in my article, cyclic rings, I would like the word "cyclic" to link with the article on cyclic groups; however, it links to the article on cyclic quadrilaterals instead.
	planetmath.org /encyclopedia/CyclicRing3.html (331 words)

	PlanetMath: quadrilateral (Site not responding. Last check: 2007-10-08)
		A very special kind of quadrilaterals are parallelograms (squares, rhombuses, rectangles, etc) although cyclic quadrilaterals are also interesting on their own.
		Notice however, that there are quadrilaterals that are neither parallelograms nor cyclic quadrilaterals.
		This is version 3 of quadrilateral, born on 2001-12-11, modified 2005-04-20.
	planetmath.org /encyclopedia/Quadrilateral.html (61 words)

	Math Forum - Ask Dr. Math
		When this quadrilateral was inscribed, I saw that the opposing angles would be supplementary, and therefore the area would be maximized because the cosine term would be 0.
		Also I know that a quadrilateral inscribed in a circle has opposing angles that add to 180 degrees, but I don't know how to show the converse, that if a quadrilateral has supplementary opposing angles, then it can be placed inside a circle.
		I once proved the cyclicity of the maximal-area quadrilateral with given side lengths by setting up the area as the sum of the areas of the two triangles formed by a diagonal, and then taking the derivative with respect to the length of that diagonal.
	mathforum.org /library/drmath/view/51795.html (752 words)

	Concyclic Points (Site not responding. Last check: 2007-10-08)
		For example, ABCD is a cyclic quadrilateral since the vertices A, B, C and D lie on the circle.
		Opposite angles of a cyclic quadrilateral are supplementary.
		The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
	www.mathsteacher.com.au /year10/ch08_further_geometry/04_cyclic_quadrilaterals/cyclic.htm (246 words)

	Heron's Formula and Brahmagupta's Generalization (Site not responding. Last check: 2007-10-08)
		This is the area of a quadrilateral with sides a,b,c,d inscribed in a circle, i.e., a cyclic quadrilateral.
		Naturally every triangle is cyclic, meaning that it can be inscribed in a circle, and a triangle can be regarded as a quadrilateral with one of its four edge lengths set equal to zero.
		Incidentally, the formula for the area of an arbitrary quadrilateral is 1 ________________________________________________________ --- /(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) - 16 abcd cos(q)^2 4 where q is half the sum of two opposite angles.
	www.mathpages.com /home/kmath196.htm (512 words)

	College Mathematics Journal, The: A property of quadrilaterals (Site not responding. Last check: 2007-10-08)
		The diagonals of a quadrilateral play a key role in many associated properties, especially with cyclic quadrilaterals, ones whose vertices lie on a circle [1].
		Cyclic quadrilaterals are also noteworthy because those are the ones of maximum area formed from four given sides.
		If we are given a convex quadrilateral ABCD, as shown in the figure, and a, b, c, d, are the lengths of the four sides, then the sum of the squares of these sides is related to the lengths of the two diagonals by
	www.findarticles.com /p/articles/mi_qa3773/is_200109/ai_n8994126 (352 words)

	EMAT 8990 (Site not responding. Last check: 2007-10-08)
		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.
		Let the diagonals of cyclic quadrilateral ABCD meet at M and let their midpoints be E and F. Then the orthocenter of triangle EMF is the anticenter T of ABCD.
	jwilson.coe.uga.edu /EMT668/EMAT6680.2000/Westmoreland/gems/cyclicquads/cyclicquads.html (286 words)

	Search Results for cyclic - Encyclopædia Britannica
		Some cyclic eliminations are fully concerted, but in others the loss of a nucleophilic or of an electrophilic component can be dominant.
		There is good evidence of cyclic secretion of substances in the brain, which appears to be related to the control of molting and reproduction.
		Guide to this cyclic solar phenomenon, expected to be at a peak in the year 2000.
	www.britannica.com /search?query=cyclic&submit=Find&source=MWTEXT (514 words)

	Cyclic - Cyclic redundancy check - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-08)
		Cyclic multivariate data is encountered in a variety of disciplines, Previously, data of this cyclic multivariate type, or cyclic univariate data for
		cyclic group In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a.
		A cyclic redundancy check (CRC) is a type of hash function used to produce a checksum, which is a small number of bits, from a large block of data,
	indexingnet.com /?q=cyclic (430 words)

	Cyclic quadrilateral -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Opposite angles are supplementary angles (adding up to either 180 in degrees or π in radians).
		The area of a cyclic quadrilateral is given by (Click link for more info and facts about Brahmagupta's formula) Brahmagupta's formula as well as (Click link for more info and facts about Heron's formula) Heron's formula as long as the sides are given.
		The product of the two diagonals is equal to the sum of the products of opposite sides.
	www.absoluteastronomy.com /encyclopedia/c/cy/cyclic_quadrilateral.htm (177 words)

	Heron Quadrilaterals with Sides in Arithmetic or Geometric Progression - Buchholz, MacDougall (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions.
		A complete characterization is given for the in nite family of triangles with sides in arithmetic progression.
		We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression.
	citeseer.ist.psu.edu /493787.html (377 words)

	Mathematics Magazine: Perfect cyclic quadrilaterals (Site not responding. Last check: 2007-10-08)
		Are there any quadrilaterals with integer sides having perimeter P equal to area A? A square of side length 4 might come to mind.
		This question is interesting mainly for cyclic quadrilaterals (that is, those that can be inscribed in a circle) since there are, for example, an infinite number of parallelograms satisfying P = kA for a given positive number k (as the reader can check).
		The purpose of this paper is to extend these results and discuss the number N(k) of cyclic quadrilaterals with integer sides (including 1) satisfying P = kA, where k is a positive real number, and P and A are the perimeter and area of a quadrilateral.
	newssearch.looksmart.com /p/articles/mi_qa3789/is_200204/ai_n9073353 (502 words)

	Collinearity in Bicentric Quadrilaterals
		Prove that in bicentric quadrilaterals the incenter I, the circumcenter O and the point of intersection E of its diagonals are collinear.
		By Brahmagupta, PK, PL, PM, and PN are maltitudes in quadrilateral ABCD, i.e., their extensions beyond P cross the opposite sides at their midpoints.
		Rectangle is a cyclic shape with the circumcenter at the intersection of the diagonals, both of which are diameters of the incircle
	www.maa.org /editorial/knot/BicentricQuadri.html (949 words)

	Cyclic Quadrilateral (@LSKCSite)
		A cyclic quadrilateral is a quadrilateral that could be inscribed in a circle, or in other words, there is a circle that circumscribes the quadrilateral.
		Note that squares and rectangles are cyclic quadrilaterals, while parallelograms which are not rectangles are not cyclic quadrilaterals.
		If ABCD is a quadrilateral with an exterior angle equal to the opposite interior angle, then it is a cyclic quadrilateral (converse of ext.
	family.lskc.edu.hk /subnotes/wakka.php?wakka=CyclicQuadrilateral (149 words)

	Quadrilaterals - Quadrilaterals Discussion
		Quadrilaterals Types of Quadrilaterals Finding the Area of a Quadrilateral Area of a Polygon Diagonals Diagram 1: Convex and Re-entrant Quadrilaterals
		The following formulas give the area of a general quadrilateral (see More formulas can be given for special cases of quadrilaterals.
		Properties Of Quadrilaterals Stephen Maraldo Definitions, theorems, and a diagram.
	findoutpages.com /?q=quadrilaterals (244 words)

	INVESTIGATING HISTORICAL PROBLEMS
		GSP is an excellent environment for one to test whether a quadrilateral is cyclic or not.
		Seeking out quadrilaterals that are cyclic provides a nice exploration of this theorem and can easily be examined using GSP (Figure 3(a)).
		Students are provided with a situation where they can make generalizations about various quadrilaterals and their unique properties and look for patterns among cyclic quadrilaterals.
	www.math.iup.edu /MAA/proceedings/vol1/enderson/enderson.htm (1394 words)

	Cyclic_Quadrilateral (Site not responding. Last check: 2007-10-08)
		Objective: The students will be able to use the Cyclic Quadrilateral Theorem to determine the measure of angles in a circle and to prove quadrilaterals are cyclic.
		That means it is possible to construct a circle through the points A, B, C, and D. Since quadrilateral BCDE is inscribed in a circle,
		2) A quadrilateral is cyclic if and only if the opposite angles are supplementary.
	mathematics.ridley.on.ca /courses/theorem/Cyclic_Quadrilateral.html (179 words)

	IMO 1972/2 solution (Site not responding. Last check: 2007-10-08)
		Given n > 4, prove that every cyclic quadrilateral can be dissected into n cyclic quadrilaterals.
		A little tinkering soon shows that it is easy to divide a cyclic quadrilateral ABCD into 4 cyclic quadrilaterals.
		For then we may take arbitarily many lines parallel to the parallel sides and divide the original quadrilateral into any number of parts.
	math.ymsh.tp.edu.tw /exams/IMO/isoln722.html (267 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-08)
		Date: 8/30/96 at 9:27:23 From: Stephen Digby Subject: Cyclic Quadrilaterals A cyclic quadrilateral with angles of 120, 60, 100, 80 touches a circle at each vertex.
		Date: 8/30/96 at 13:50:58 From: Doctor Tom Subject: Re: Cyclic Quadrilaterals Any angle inscribed in a circle cuts off an arc of twice its size, so the arcs have sizes 240, 120, 200, and 160 degrees.
		Draw a picture and draw a heavy line over the indicated arcs to convince yourself that they completely cover the circle twice, so a check of this answer should be that the angles should add to 2 times 360 or 720 degrees.
	mathforum.org /library/drmath/view/54877.html (190 words)

	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.
		In a cyclic quadrilateral having perpendicular diagonals, the perpendicular from the midpoint of a side to the opposite side passes through the point of intersection of the diagonals.
		There are four such perpendiculars and all four pass through the point of intersection of the diagonals.
	www.cut-the-knot.org /Curriculum/Geometry/Brahmagupta.shtml (218 words)

	RHS - Cyclic Quadrilaterals (Site not responding. Last check: 2007-10-08)
		This site will show you everything you need to know about cyclic quadrilaterals.
		A cyclic quadrilateral is a quadrilateral where a single circle passes through all four vertices.
		The quadrilateral is inscribed into the circle, or the circle circumscribes the quadrilateral.
	grassroots.brunnet.net /rothesayhigh/geometry/circle_geometry-cyclic_quadrilateral (41 words)

	Four Concurrent Lines in a Cyclic Quadrilateral
		Point T of concurrency is known as the anticenter of the quadrilateral ABCD.
		Let the extensions of the opposite sides AD and BC of the cyclic quadrilateral meet at a point U. Prove that UT, where T is the anticenter of ABCD, is perpendicular to SQ.
		be reflections of the circumcenter of the cyclic quadrilateral in AB and CD, respectively.
	www.cut-the-knot.com /Curriculum/Geometry/Brahmagupta2.shtml (321 words)

	Mathpuzzle.com
		I'm fairly certain that this is the smallest quadrilateral with 6 different integers for the sides and diagonals.
		What are the possible quadrilaterals, and how are they marked?" Answer.
		A cyclic cover starts with a polygon where the sides and diagonals all have different length.
	www.mathpuzzle.com /21oct02.html (2213 words)

	Geometer's Sketchpad Assignment #2
		Those quadrilaterals that can be circumscribed are called cyclic.
		Suppose Quadrilateral BCDE is a cyclic quadrilateral circumscribed by the circle centered at point A. Use Geometer's Sketchpad to draw a general cyclic quadrilateral with vertices labeled B, C, D and E. Now reflect the center of the circle (call it point A) over each of the sides of quadrilateral BCDE.
		In other words, drag the vertices of the orginal cyclic quadrilateral a few times to be sure the conjecture holds for a general cyclic quadrilateral.
	www.math.ilstu.edu /~smccrone/MAT211_F02/GSP2.html (365 words)

	Area of Triangles and Polygons (2D & 3D)
		It is then easy to show that this midpoint quadrilateral is always a parallelogram, called the "Varignon parallelogram", and that its area is exactly one-half the area of the original quadrilateral [Coxeter,1967, Section 3.1].
		For simple quadrilaterals, the area is positive when the vertices are oriented counterclockwise, and negative when they are clockwise.
		However, it also works for nonsimple quadrilaterals and is equal to the difference in area of the two regions the quadrilateral bounds.
	www.geometryalgorithms.com /Archive/algorithm_0101/algorithm_0101.htm (2621 words)

	Topic: cyclic polygon (Site not responding. Last check: 2007-10-08)
		All triangles, all rectangles, and all regular polygons are cyclic.
		Convex quadrilaterals, whose opposite angles are supplementary, are also cyclic.
		Related Terms: circle, cyclic quadrilateral, polygon, vertex (in plane geometry)
	www.elko.k12.nv.us /webapps/vmd/full/c/cyclicpolygon.htm (40 words)

	Carnot's Theorem
		The proof is based on essentially the same property of the configuration as in the Wallace's Theorem, viz.
		Taking two at a time, the quadrilaterals share an inscribed angle which in each of the quadrilaterals equals another angle.
		From here we obtain pairs of similar triangles and the corresponding side ratios, the product of which is shown to be 1.
	www.cut-the-knot.org /Curriculum/Geometry/Carnot.shtml (188 words)

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