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Topic: Circumcircle

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	Encyclopedia: Circumcircle (Site not responding. Last check: 2007-11-07)
		A polygon whose vertices all lie on its circumcircle is said to be a cyclic polygon.
		The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle.
		Circumcircles of triangles have an intimate relationship with the Delaunay triangularization of a set of points.
	www.nationmaster.com /encyclopedia/Circumcircle (1249 words)

	Circumcircle -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		The center of this circumcircle is known as the shape's circumcenter.
		The circumcircle of a triangle is the unique circle on which all its three vertices lie.
		The useful minimum bounding circle of three points is defined either by the circumcircle (where three points are on the minimum bounding circle) or by the two points of the longest side of the triangle (where the two points define a diameter of the circle.).
	www.absoluteastronomy.com /encyclopedia/c/ci/circumcircle.htm (524 words)

	PlanetMath: circumcircle (Site not responding. Last check: 2007-11-07)
		is any polygon, its circumcircle would be a circle passing throught all vertices, and circumradius and circumcenter are defined similarly.
		For instance, a non-rectangular parallelogram has no circumcircle, for no circle passes through the four vertices.
		This is version 1 of circumcircle, born on 2005-02-05.
	planetmath.org /encyclopedia/Circumcircle.html (151 words)

	Circumcircle
		Some (but not necessarily all) vertices of the shape will lie on the circumcircle.
		In particular, a polygon whose vertices all lie on its circumcircle is said to be a cyclic polygon.
		The center of a triangle's circumcircle, also called its cirumscribed circle, is called its circumcenter.
	www.ebroadcast.com.au /lookup/encyclopedia/ci/Circumcircle.html (88 words)

	Tim's Triangular Page (Site not responding. Last check: 2007-11-07)
		The center of the circumcircle is located at the point of intersection of three lines perpendicular to triangle sides and crossing them at the midpoints.
		The center of the circumcircle of acute triangles is located inside the triangles whereas it is located outside of obtuse triangles.
		The radius of the incircle is BE/3, the radius of the circumcircle is 2*BE/3.
	sakharov.net /triangle.html (3567 words)

	K003
		Locus of point P such that the polar lines (in the circumcircle) of P and its isogonal conjugate P* are parallel.
		Locus of point P such that the antipedal triangle of P and the circumcevian triangle of its isogonal conjugate are perspective (together with the line at infinity and the circumcircle).
		More generally, when P and Q are two isoconjugates under an isoconjugation with pole W, the locus is a member of the class CL021 of cubics (together with the line at infinity, its W-isoconjugate and the trilinear polar of the isotomic conjugate of the isogonal conjugate of W).
	perso.wanadoo.fr /bernard.gibert/Exemples/k003.html (977 words)

	Simson line (Site not responding. Last check: 2007-11-07)
		Similarly, find the reflection of BP in the bisector of B and that of CP in the bisector of C. It's a curious fact that the three lines thus obtained are parallel.
		The feet of the perpendiculars from points outside the circumcircle to the sides of the triangle are not colinear.
		As P moves over the circumcircle, the point of intersection of the two lines traces a circle, the 9-point circle in the case of the (regular) Simson lines.
	www.maa.org /editorial/knot/SimsonLine.html (1119 words)

	Orthocenter
		One thing that may not be immedaitely obvious to the reader, is that if the Point A was moved around the circle holding B and C stationary, the orthocenter would follow the path of this circle.
		Well, an easy answer is that we know know that RQ is a chord of the circumcircle, and that the line from A perpendicular to RQ must not only bisect RQ, but pass through the center of the circle.
		Stated another way, a radius of the circumcircle to a vertex of ABC is perpendicular to the side of the double-orthic triangle.
	www.pballew.net /orthocen.html (1819 words)

	Circumcircle (Site not responding. Last check: 2007-11-07)
		The circumcircle of a triangle is a circle that passes through all of the vertices of the triangle.
		The circumcircle of a polygon is a circle that passes through all of the vertices of the polygon.
		The circumcenter of a triangle is the center of the circumcircle of the triangle.
	illumtest.nctm.org /mathlets/IGD_lines/Circumcircle.html (372 words)

	[No title]
		And while the Schröder point Sc is the inverse of X(55), the internal center of similtude of circumcircle and incircle, in the circumcircle of ABC, the Bevan-Schröder point Sb is the inverse of X(56), the external center of similtude of circumcircle and incircle, in the circumcircle of ABC.
		The internal and external centers of similtude of the circumcircle and the incircle are poristically fixed.
		By inversion in the circumcircle, this yields that the Schröder point and the Bevan-Schröder point are poristically fixed (what means that all triangles which share the same circumcircle and the same incircle have the same Schröder point and the same Bevan-Schröder point).
	de.geocities.com /darij_grinberg/Schroeder/Schroeder.html (6708 words)

	Circumcircle - Encyclopedia, History, Geography and Biography
		At least two vertices of a shape will lie on its circumcircle.
		v^2 and v_x and v_y and 1 \\ A^2 and A_x and A_y and 1 \\ B^2 and B_x and B_y and 1 \\ C^2 and C_x and C_y and 1 \end{vmatrix}=0 where A, B and C are the points of the triangle, and the solution for v is the circumcircle.
		The article about Circumcircle contains information related to Circumcircle, Cyclic polygons, Circumcircles of triangles, Circumcircle equation, Circumcircle of circles, See also and External links.
	www.arikah.net /encyclopedia/Circumcenter (581 words)

	ENCYCLOPEDIA OF TRIANGLE CENTERS - PART 5
		As the isogonal conjugate of a point on the circumcircle, X(802) lies on the line at infinity.
		As the isogonal conjugate of a point on the circumcircle, X(804) lies on the line at infinity.
		As the isogonal conjugate of a point that lies on the circumcircle, X(900) lies on the line at infinity.
	faculty.evansville.edu /ck6/encyclopedia/part5.html (6251 words)

	ENCYCLOPEDIA OF TRIANGLE CENTERS - PART 4
		As the isogonal conjugate of a point on the circumcircle, X(700) lies on the line at infinity.
		As the isogonal conjugate of a point on the circumcircle, X(702) lies on the line at infinity.
		[For nonzero n, "odd (m,n) circumcircle point" is would be a misnomer (as the point is an even polynomial center); consequently, the prefix o- is used to distinguish this point from "even (m,n) circumcircle point" defined at X(696).] Certain points of these classes occur prior to this section.
	faculty.evansville.edu /ck6/encyclopedia/part4.html (7727 words)

	standard
		The Gergonne and Nagel points are on the Feuerbach hyperbola since their isogonal conjugates are the centers of similitudes of the circumcircle and the incircle.
		The isodynamic points X(15), X(16) are the intersection of the Parry circle (2,110,111) and the Brocard axis OK. The two points are inverses with respect to the circumcircle, and X(15) is the interior one.
		The giving of the sidelengths of triangle OHK is equivalent with giving R (radius of circumcircle), DELTA (area of triangle ABC) and Q = a^2 + b^2 + c^2 (where a, b and c are the sidelengths of triangle ABC).
	forumgeom.fau.edu /POLYA/ProblemCenter/POLYA022.html (1836 words)

	Count On (Site not responding. Last check: 2007-11-07)
		Point P is on the circumcircle and R is its reflection in AB, S in AC and T in BC.
		As the point moves on the circumcircle, so the line rotates but it always passes through the orthocentre of the triangle.
		If the two results are combined, so that the image line of the point Q is created, then amazingly this line goes through the starting point P. Click here to open the combined reflections java applet in a separate window.
	www.mathsyear2000.org /explorer/circles/triangles-and-reflections (402 words)

	K004
		Locus of point P such that the pedal and antipedal triangles of P are perspective (together with the line at infinity and the circumcircle).
		Locus of point P such that the six vertices of the pedal and cevian (or antipedal and anticevian) triangles of P lie on a same conic.
		(Ha) is the hyperbola centered at A, passing through the antipode Ao of A in the circumcircle, through its reflection A1 in A and whose asymptotes are the perpendiculars at A to the sidelines AB and AC (these are the lines ABo and ACo).
	perso.wanadoo.fr /bernard.gibert/Exemples/k004.html (700 words)

	InterMath / Dictionary / Description
		Circumcircle: The circle that passes through all the vertices of a polygon.
		The figures below show circumcircles (in red) of regular polygons.
		The polygons do not have to be regular, but they do need to be convex, as the following examples illustrate.
	www.intermath-uga.gatech.edu /dictnary/descript.asp?termID=69 (65 words)

	Circumcircle (Site not responding. Last check: 2007-11-07)
		A circle that passes through each of the vertices of a polygon is called the circumcircle of the polygon.The center of that circle is the circumcenter.Both seem to be truncations of the prefix “circumscribing” attached to the words circle or center.The word circumcircle was suggested in 1883 by W. Hudson in Nature Magazine, according to
		Because the circumcenter must be the same distance from each of the vertices, and because each side of the polygon is a chord of the circle, it is easy to understand that the circumcircle must be at the intersection of the perpendicular bisectors of the sides of the polygon.
		Steiner's theorem, pictured at right, gives a relationship between the radii of five important circles in any triangle, the cirucmcircle, the incircle, and the three excircles (circles tangent to the three sides (or sides extended) of a triangle on the outside of the triangle.
	www.pballew.net /Circumcircle.htm (396 words)

	klein view page
		There are many points associated with a triangle, such as the incentre and circumcentre.
		These are all the named centers known to lie on the circle.
		Also, inversion in a circle with centre X(15) or X(16) maps the circumcircle and Brocard circle to concentric circles.
	www.maths.gla.ac.uk /~wws/cabripages/misc/misc0.html (872 words)

	MathLinks Math Forum :: View topic - nice and hard as usual
		These lines pass through 2 certain fixed points on the image of the circumcircle, which are the images of the reflections of H in CA, AB by the given inversion (call these fixed images Xb and Xc).
		Therefore, they have two common points (namely the point H and their second point of concurrence on the circumcircle of triangle ABC), meaning that they are coaxal.
		I tried to prove that the circumcircles of HA1A2 and HB1B2 and HC1C2 have another common point beyond H.(its obvious that this is the same as what the problem mentions).Then i inverted through H with the power HA.HA''(A'' is the foot of the altitude from A).
	www.mathlinks.ro /Forum/post-20276.html (3025 words)

	More Advanced Facts about Triangles (Site not responding. Last check: 2007-11-07)
		In any triangle ABC, the distance from the circumcircle center to BC is half of the distance from the orthocenter to A. The radius of the circumcircle is less than half of the radius of the incircle for any triangle.
		If c is the radius of the circumcircle, i is the radius of the incircle of one triangele, and the distance between the centers of these circles is d, then d
		The distance from a triangle vertex to the orthocenter is twice the distance from the circumcircle center to the opposite side.
	sakharov.net /trianglemore.html (474 words)

	3.2.4 -Delaunay triangulation (Site not responding. Last check: 2007-11-07)
		However, the most useful method is to generate the points simultaneously with the triangulation, choosing the points to improve the quality of the triangles [30].
		Quality is defined by (e.g.) size of circumcircle relative to the locally required value interpolated from a background grid.
		In a constrained Delaunay triangulation the pre-defined edges are in the triangulation and the empty circumcircle property is modified to apply only to points that can be `seen' from at least one node of the triangle, where the pre-defined edges are treated as opaque.
	www.epcc.ed.ac.uk /overview/publications/training_material/tech_watch/96_tw/tw-meshgen/MeshGeneration.book_29.html (1259 words)

	Definitions (Site not responding. Last check: 2007-11-07)
		A circumcircle is any one of the circles in a set of circles satisfying the empty circumcircle criterion.
		A precise definition can be given in terms of the empty circumcircle criterion.
		Any two points in a finite subset of the plane are said to be natural neighbors if they lie on the same circumcircle.
	ngwww.ucar.edu /ngdoc/ng/ngmath/definitions.html (1354 words)

	Phillipe's remarks (Site not responding. Last check: 2007-11-07)
		You are correct that the circumCircle method would be difficult for non-finite geometries.
		An option would be to take circumCircle out of the base class and only define it in classes of finite geometries.
		You are correct that the circumCircle > method would be difficult for non-finite > geometries.
	www.indiana.edu /~aem/oo-aem_bbs/oo-aem.cgi?read=30 (534 words)

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