|
|
| Line 1: |
Line 1: |
| [[Image:Concyclic.svg|thumb|right|250px|right|[[Concurrent lines|Concurrent]] perpendicular bisectors of [[Circle#Chord|chords]] between concyclic points]]
| | Roberto is the name Simply put i love to be titled with though I commonly do not really like being which is called like that. My pals say it's not superb for me but methods I love doing is definitely to bake but I'm just thinking on starting something totally new. South Carolina is where some of my home is. Software developing is how As well as support my family. You can consider my website here: http://[http://www.twitpic.com/tag/prometeu.net prometeu.net]<br><br> |
| [[Image:Four concyclic points.png|thumb|right|250px|right|Four concyclic points forming a [[cyclic quadrilateral]], showing two equal angles]]
| |
| In [[geometry]], a [[set (mathematics)|set]] of [[point (geometry)|points]] are said to be '''concyclic''' (or cocyclic) if they lie on a common [[circle]].
| |
|
| |
|
| ==Bisectors==
| | Here is my blog post :: [http://prometeu.net clash of clans hack] |
| In general the centre ''O'' of a circle on which points ''P'' and ''Q'' lie must be such that ''OP'' and ''OQ'' are equal distances. Therefore ''O'' must lie on the [[perpendicular bisector]] of the [[line segment]] ''PQ''.<ref>{{citation
| |
| | last = Libeskind | first = Shlomo
| |
| | isbn = 9780763743666
| |
| | page = 21
| |
| | publisher = Jones & Bartlett Learning
| |
| | title = Euclidean and Transformational Geometry: A Deductive Inquiry: A Deductive Inquiry
| |
| | url = http://books.google.com/books?id=6YUUeO-RjU0C&pg=PA21
| |
| | year = 2008}}/</ref> For ''n'' distinct points there are [[triangular number|''n''(''n'' − 1)/2]] bisectors, and the concyclic condition is that they all meet in a single point, the centre ''O''.
| |
| | |
| ==Cyclic polygons==
| |
| The vertices of every [[triangle]] fall on a circle. (Because of this, some authors define "concyclic" only in the context of four or more points on a circle.)<ref>{{citation
| |
| | last = Elliott | first = John
| |
| | page = 126
| |
| | publisher = Swan Sonnenschein & co.
| |
| | title = Elementary Geometry
| |
| | url = http://books.google.com/books?id=9psBAAAAYAAJ&pg=PA126
| |
| | year = 1902}}.</ref> The circle containing the vertices of a triangle is called the [[circumscribed circle]] of the triangle. Several other sets of points defined from a triangle are also concyclic, with different circles; see [[nine-point circle]]<ref>{{citation
| |
| | last = Isaacs | first = I. Martin
| |
| | isbn = 9780821847947
| |
| | page = 63
| |
| | publisher = American Mathematical Society
| |
| | series = Pure and Applied Undergraduate Texts
| |
| | title = Geometry for College Students
| |
| | url = http://books.google.com/books?id=0ahK8UneO3kC&pg=PA63
| |
| | volume = 8
| |
| | year = 2009}}.</ref> and [[Lester's theorem]].<ref>{{citation
| |
| | last = Yiu | first = Paul
| |
| | journal = Forum Geometricorum
| |
| | mr = 2868943
| |
| | pages = 175–209
| |
| | title = The circles of Lester, Evans, Parry, and their generalizations
| |
| | url = http://forumgeom.fau.edu/FG2010volume10/FG201020.pdf
| |
| | volume = 10
| |
| | year = 2010}}.</ref>
| |
| | |
| The radius of the circle on which lie a set of points is, by definition, the radius of the circumcircle of any triangle with vertices at any three of those points. If the pairwise distances among three of the points are ''a'', ''b'', and ''c'', then the circle's radius is
| |
| | |
| :<math>R = \sqrt{\frac{a^2b^2c^2}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}.</math>
| |
| | |
| The equation of the circumcircle of a triangle, and expressions for the radius and the coordinates of the circle's center, in terms of the Cartesian coordinates of the vertices are given [[Circumscribed circle#Circumcircle equations|here]] and [[Circumscribed circle#Cartesian coordinates|here]].
| |
| | |
| A quadrilateral ''ABCD'' with concyclic vertices is called a [[cyclic quadrilateral]]; this happens if and only if <math>\angle CAD = \angle CBD</math> (the [[inscribed angle theorem]]) which is true if and only if the opposite angles inside the quadrilateral are [[supplementary angle|supplementary]].<ref>{{citation
| |
| | last = Pedoe | first = Dan
| |
| | edition = 2nd
| |
| | isbn = 9780883855188
| |
| | page = xxii
| |
| | publisher = Cambridge University Press
| |
| | series = MAA Spectrum
| |
| | title = Circles: A Mathematical View
| |
| | url = http://books.google.com/books?id=rlbQTxbutA4C&pg=PR22
| |
| | year = 1997}}.</ref> A cyclic quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' and [[semiperimeter]] ''s''= (''a''+''b''+''c''+''d'')/2 has its circumradius given by<ref name=Alsina2>{{citation
| |
| |last1=Alsina |first1=Claudi |last2=Nelsen |first2=Roger B.
| |
| |journal=Forum Geometricorum
| |
| |pages=147–9
| |
| |title=On the diagonals of a cyclic quadrilateral
| |
| |url=http://forumgeom.fau.edu/FG2007volume7/FG200720.pdf |format=PDF
| |
| |volume=7
| |
| |year=2007}}</ref><ref>{{citation |last=Hoehn |first=Larry |title=Circumradius of a cyclic quadrilateral |journal=Mathematical Gazette |volume=84 |issue=499 |date=March 2000 |pages=69–70 |jstor=3621477}}</ref>
| |
| :<math>R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}},</math>
| |
| an expression that was derived by the Indian mathematician Vatasseri [[Parameshvara]] in the 15th century.
| |
| | |
| By [[Ptolemy's theorem]], if a quadrilateral is given by the pairwise distances between its four vertices ''A'', ''B'', ''C'', and ''D'' in order, then it is cyclic if and only if the product of the diagonals equals the sum of the products of opposite sides:
| |
| | |
| : <math>AC \cdot BD = AB \cdot CD + BC \cdot AD.</math>
| |
| | |
| If two lines, one containing segment ''AC'' and the other containing segment ''BD'', intersect at ''X'', then the four points ''A'', ''B'', ''C'', ''D'' are concyclic if and only if<ref>{{citation |last=Bradley |first=Christopher J. |title=The Algebra of Geometry: Cartesian, Areal and Projective Co-Ordinates |publisher=Highperception |year=2007 |isbn=1906338000 |page=179 |oclc=213434422}}</ref>
| |
| :<math>\displaystyle AX\cdot XC = BX\cdot XD.</math>
| |
| | |
| The intersection ''X'' may be internal or external to the circle.
| |
| | |
| More generally, a polygon in which all vertices are concyclic is called a [[cyclic polygon]]. A polygon is cyclic if and only if the perpendicular bisectors of its edges are concurrent.<ref>{{citation
| |
| | last1 = Byer | first1 = Owen
| |
| | last2 = Lazebnik | first2 = Felix
| |
| | last3 = Smeltzer | first3 = Deirdre L.
| |
| | isbn = 9780883857632
| |
| | page = 77
| |
| | publisher = Mathematical Association of America
| |
| | title = Methods for Euclidean Geometry
| |
| | url = http://books.google.com/books?id=W4acIu4qZvoC&pg=PA77
| |
| | year = 2010}}.</ref>
| |
| | |
| ==Variations==
| |
| Some authors consider [[collinear points]] (sets of points all belonging to a single line) to be a special case of concyclic points, with the line being viewed as a circle of infinite radius. This point of view is helpful, for instance, when studying [[Inversive geometry|inversion through a circle]] and [[Möbius transformation]]s, as these transformations preserve the concyclicity of points only in this extended sense.<ref>{{citation
| |
| | last = Zwikker | first = C.
| |
| | isbn = 9780486442761
| |
| | page = 24
| |
| | publisher = Courier Dover Publications
| |
| | title = The Advanced Geometry of Plane Curves and Their Applications
| |
| | url = http://books.google.com/books?id=25tMEYTik-AC&pg=PA24
| |
| | year = 2005}}.</ref>
| |
| | |
| In the [[complex plane]] (formed by viewing the real and imaginary parts of a [[complex number]] as the ''x'' and ''y'' [[Cartesian coordinates]] of the plane), concyclicity has a particularly simple formulation: four points in the complex plane are either concyclic or collinear if and only if their [[cross-ratio]] is a [[real number]].<ref>{{citation
| |
| | last = Hahn | first = Liang-shin
| |
| | edition = 2nd
| |
| | isbn = 9780883855102
| |
| | page = 65
| |
| | publisher = Cambridge University Press
| |
| | series = MAA Spectrum
| |
| | title = Complex Numbers and Geometry
| |
| | url = http://books.google.com/books?id=s3nMMkPEvqoC&pg=PA65
| |
| | year = 1996}}.</ref>
| |
| | |
| ==Other properties==
| |
| A set of five or more points is concyclic if and only if every four-point subset is concyclic.<ref>{{citation
| |
| | last = Pedoe | first = Dan
| |
| | isbn = 9780486658124
| |
| | page = 431
| |
| | publisher = Courier Dover Publications
| |
| | title = Geometry: A Comprehensive Course
| |
| | url = http://books.google.com/books?id=s7DDxuoNr_0C&pg=PA431
| |
| | year = 1988}}.</ref> This property can be thought of as an analogue for concyclicity of the [[Helly property]] of convex sets.
| |
| | |
| [[Abouabdillah's theorem]] characterizes the [[similarity transformation (geometry)|similarity transformation]]s of a [[Euclidean space]] of dimension two or more as being the only [[surjective]] mappings of the space to itself that preserve concyclicity.<ref>{{citation|first=D.|last=Abouabdillah|title=Sur les similitudes d'un espace euclidien|journal=Revue de Mathématiques Spéciales|volume=7|year=1991}}.</ref>
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| == External links ==
| |
| * {{MathWorld |title=Concyclic |urlname=Concyclic}}
| |
| *''[http://demonstrations.wolfram.com/FourConcyclicPoints/ Four Concyclic Points]'' by Michael Schreiber, [[The Wolfram Demonstrations Project]].
| |
| | |
| [[Category:Elementary geometry]]
| |
Roberto is the name Simply put i love to be titled with though I commonly do not really like being which is called like that. My pals say it's not superb for me but methods I love doing is definitely to bake but I'm just thinking on starting something totally new. South Carolina is where some of my home is. Software developing is how As well as support my family. You can consider my website here: http://prometeu.net
Here is my blog post :: clash of clans hack