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Topic: Circumcenter

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	Altitudes, Orthocenters, and Scripting (Site not responding. Last check: 2007-10-08)
		2) The orthocenter of a triangle is the circumcenter of its anticomplementary triangle.
		3) The circumcenter of a triangle is the orthocenter of its complementary triangle.
		Furthermore, the distance between the centroid and the orthocenter is twice the distance between the centroid and the circumcenter.
	www.beva.org /math323/asgn4/oct22.htm (205 words)

	All about altitudes
		Orthocenter as the Isogonal Conjugate of the Circumcenter
		The same is true for the vertices B and C. Therefore, H is the isogonal conjugate of the circumcenter O.
		The argument that shows that three points - the circumcenter O, the centroid M, and the orthocenter H - lie on the same line is reversible.
	www.cut-the-knot.org /triangle/altitudes.shtml (1300 words)

	circumcenter (Site not responding. Last check: 2007-10-08)
		The circumcenter of a triangle is the point of intersection of the three perpendicular bisectors of a triangle.
		If the triangle is a right triangle (one of the angles of the triangle is equal to 90 degrees), then the circumcenter will lie on the midpoint of the hypotenuse of the triangle.
		For triangle XYZ, X', Y', and Z' are the midpoints of the sides and C is the circumcenter of the triangle.
	www.auburn.edu /~orazikl/ctse4040/circumcenter.htm (206 words)

	Triangle -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.
		The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine point circle (red point) all lie on a single line, known as (additional info and facts about Euler's line) Euler's line (red line).
		The center of the nine point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
	www.absoluteastronomy.com /encyclopedia/t/tr/triangle.htm (2236 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		% % Discussion: % % The circumcenter of a triangle is the center of the circumcircle, the % circle that passes through the three vertices of the triangle.
		% % The circumcenter is the intersection of the perpendicular bisectors % of the sides of the triangle.
		% % In geometry, the circumcenter of a triangle is often symbolized by "O".
	www.csit.fsu.edu /~burkardt/m_src/geometry/triangle_circumcenter_2d_2.m (373 words)

	Untitled Document (Site not responding. Last check: 2007-10-08)
		The circumcenter of a triangle is the point equidistant from the three vertices.
		Note the circumcenter is inside an acute triangle, on the side of a right triangle, and outside an obtuse triangle.
		The circumcenter is the intersection of the three perpendicular bisectors.
	jwilson.coe.uga.edu /emt668/EMAT6680.2000/Lehman/assign4/assign4.html (229 words)

	Where are the Centers? (Site not responding. Last check: 2007-10-08)
		For a right triangle, the Orthocenter is on the right angle vertex, the Circumcenter is the midpoint of hypotenuse, and the other 2 centers are somewhere inside the triangle.
		An additional interesting fact: we should not be surprised to see that the Circumcenter falls on the midpoint of the hypotenuse, because the midpoint of the hypotenuse of a right triangle is equidistant from the 3 vertices of the triangle.
		In an obtuse triangle, the Circumcenter and Orthocenter are outside the triangle, while the other 2 centers are inside the triangle.
	www.punahou.edu /acad/sanders/CenterWhere.html (533 words)

	Circumcenter Construction (Site not responding. Last check: 2007-10-08)
		This is how you construct a circumcenter and circumscribed circle in a triangle.
		The intersection of the three is the circumcenter.
		After the circumcenter is constructed, one can complete the circumscribed circle by using the distance from the intersection to a vertex as the radius.
	constructions.homestead.com /files/circ/cons7.htm (53 words)

	InterMath / Dictionary / Description
		Circumcenter: The point of intersection of the perpendicular bisectors of the sides of a given triangle; the center of the circle circumscribed about a given triangle.
		The following figure shows how the circumcenter of a triangle can be constructed - by constructing the perpendicular bisectors of the sides.
		Here is a picture of the circumcenter and circumcircle of that same triangle:
	www.intermath-uga.gatech.edu /dictnary/descript.asp?termID=402 (61 words)

	Various points related to triangles
		These include the centroid, the circumcenter, the orthocenter, the incenter, the excenters, and the Euler line (which is a line, rather than a point-- can you trust anything I say?).
		The circumcenter is also the center of the circle passing through the three vertices, which circumscribes the triangle.
		The Euler line of a triangle is the line that passes through the orthocenter, the circumcenter, and the centroid.
	www.math.sunysb.edu /~scott/mat360.spr04/cindy/various.html (359 words)

	Homework 1 (Site not responding. Last check: 2007-10-08)
		The circumcenter is found by extending a perpendicular line from the midpoint of each side.
		Therefore, the circumcenter is at an equal distance (the radius) from each individual vertex.
		The distance from the circumcenter to the centroid is half the distance from the orthocenter to the centroid.
	www.math.uga.edu /~clint/2004/5200/jeffhall.htm (517 words)

	Remarkable Line in Cyclic Quadrilateral
		In a triangle, the orthocenter H, the 9-point center N, the centroid G, and the circumcenter O are collinear.
		If the quadrilateral is cyclic, all four triangles share the circumcircle and the circumcenter O. In particular, in that case, the four Euler lines concur at the common circumcenter of the four triangles.
		For the second approach, let's put the circumcenter O at the origin, and represent the vertices A, B, C, D of the quadrilateral by unit complex numbers a, b, c, d.
	www.cut-the-knot.org /Curriculum/Geometry/InscribedQuadri.shtml (625 words)

	The Complete Quadrilateral (Site not responding. Last check: 2007-10-08)
		As in the problem statement, I shall consider the three most important: the centroid, the circumcenter and the orthocenter of a triangle.
		For the centroid, the circumcenter and the orthocenter it is quite clear that if the lines intersect on BD for one position of C, then the same is true for all other positions as well.
		ABD drawn from C. For the circumcenter and the orthocenter C could be taken to be the foot of the altitude from A. Then both
	www.maa.org /editorial/knot/CompleteQuadrilateral.html (1548 words)

	Teaching Plan for Geometer's Sketchpad (Site not responding. Last check: 2007-10-08)
		This lesson plan is to introduce the concepts of circumcenter by using computers with sketchpad software to explore.
		In a triangle ABC, suppose that O is the point of circumcenter of triangle ABC.
		Suppose O is the point of circumcenter of triangle DEF, and the angle DEF is 130 degrees, then the angle DOF is ________ degrees.
	www.mste.uiuc.edu /courses/ci407su01/students/south/ychen17/termproject/gsplesson/teachplan3.html (673 words)

	Vector – Circumcenter and Orthocenter of Triangle (Site not responding. Last check: 2007-10-08)
		The representation of circumcenter or orthocenter of a triangle by using its three vertices is quite complex.
		With this in mind, we just take a look at circumcenter and orthocenter of a triangle in 2-dimensional system.
		Circumcenter can be found via the intersection of two perpendicular bisectors of the sides of a triangle.
	www.scienceoxygen.com /mathnote/vector205.html (315 words)

	Circumcircle - TheBestLinks.com - Circumcenter, Angle, Circle, Geometry, ... (Site not responding. Last check: 2007-10-08)
		Circumcenter, Circumcircle, Angle, Circle, Geometry, Iff, Polygon, Sine, Vertex...
		Some (but not necessarily all) vertices of the shape will lie on the circumcircle.
		The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors.
	www.thebestlinks.com /Circumcenter.html (442 words)

	Euler Segment
		Notice that not only is the centroid always between the circumcenter and the orthocenter, it is half as far from the circumcenter as from the orthocenter.
		If you know about similar triangles and their properties, try to show that the point on the line joining the circumcenter and centroid, twice as far from the centroid as the circumcenter and on the opposite side is the othocenter.
		The Circumcenter of a triangle is the intersection of the perpendicular bisectors (shown in blue-green) of its sides.
	www.eshu.net /elements/EG.html (405 words)

	Math Online Q and A: Please give me some real world applications of the (Site not responding. Last check: 2007-10-08)
		The circumcenter is formed be the perpendicualr bisectors of a triangle.
		Put another way, the circumcenter is a point equidistant from the three points of a triangle.
		The point of intersection will be the circumcenter of the triangle formed by the three points and since the three points ar part of the original arc or circle, the circumcenter is also the center.
	www.mathonline.org /mho/view.cgi?16688 (494 words)

	AoPS Math Forum :: View topic - Circumcenters pass through common point
		For every triangle ABC with orthocenter H, the circumcenter of triangle BCH is the reflection of the circumcenter of triangle ABC in the line BC.
		Hence, the circumcenter of triangle BCA' coincides with the circumcenter of triangle ABC.
		Applied to triangle XYZ, Lemma 1 yields that the circumcenter F of triangle XYH' is the reflection of I in the line XY.
	www.artofproblemsolving.com /Forum/post-18194.html (682 words)

	The Circumcenter of a Triangle (Site not responding. Last check: 2007-10-08)
		The circumcenter is the point at which the three perpendicular bisectors of the sides of a triangle intersect.
		The definition of the perpendicular bisector of a side of a triangle is a line segment that is both perpendicular to a side of a triangle and passes through its midpoint.
		Use Geometer's Sketchpad and state a conjecture concerning possible locations of the circumcenter.
	www.bsu.edu /web/mdlade/Finallesson/circumcenter.html (84 words)

	Circumcircle (Site not responding. Last check: 2007-10-08)
		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.
		Several important theorems about the circumcenter of a triangle and its properties are given below.
		Carnot’s Theorem states that in any triangle, the sum of the distances from the circumcenter to the three sides of the triangle will equal the sum of the radius of the incircle plus the radius of the circumcircle.
	www.pballew.net /Circumcircle.htm (396 words)

	Nine Point Circle and Euler Line
		The Euler line is the line containing the orthocenter, centroid, and circumcenter, named after Leonhard Euler (1707-1783).
		State a conjecture concerning the relative positions of the orthocenter, circumcenter, and center of the nine-point circle.
		State a conjecture concerning the radius of the circumcirle and the radius of the nine-point circle.
	www.bsu.edu /web/mdlade/Finallesson/lesson.htm (506 words)

	Solutions to Examination 1
		A circumcenter is equidistant from the vertices of its triangle, so the circumcenters of abc and abd both lie on the bisector of edge ab.
		It follows that the circumcenter of abc is lower than the circumcenter of any non-Delaunay triangle atop ab that might be placed in the priority queue Q.
		Then both of t's lower edges are created before the sweepline passes over t's circumcenter, because each of t's lower edges is either an input edge, or an edge of a Delaunay triangle with a lower circumcenter.
	www.cs.berkeley.edu /~jrs/meshf99/exam1/sol1.html (2246 words)

	[No title]
		For completeness, here are stable expressions for the circumradius and circumcenter of a triangle, in R^2 and in R^3.
		/ / / /**************************************************************************/ void tricircumcenter(a, b, c, circumcenter*, xi, eta) double a[2]; double b[2]; double c[2]; double circumcenter[2]; double xi; double eta; { double xba, yba, xca, yca; double balength, calength; double denominator; double xcirca, ycirca; / Use coordinates relative to point `a' of the triangle.
		/ / / /**************************************************************************/ void tricircumcenter3d(a, b, c, circumcenter*, xi, eta) double a[3]; double b[3]; double c[3]; double circumcenter[3]; double xi; double eta; { double xba, yba, zba, xca, yca, zca; double balength, calength; double xcrossbc, ycrossbc, zcrossbc; double denominator; double xcirca, ycirca, zcirca; / Use coordinates relative to point `a' of the triangle.
	www.ics.uci.edu /~eppstein/junkyard/circumcenter.html (2857 words)

	Concyclic Circumcenters: A Dynamic View (Site not responding. Last check: 2007-10-08)
		It's obvious that as the points M trace straight lines, so do the circumcenters of the six triangles and so does the center of the circle on which all of them lie.
		ABC and the circumcenter of triangle S. Since S is homothetic to the antipedal triangle of G with G as the center of homothety, the above line passes through the circumcenter of the antipedal triangle.
		The Lemoine point is usually denoted by K. To sum up, the center of the circle that passes through the six circumcenters lies on the line that joins the Lemoine point K with the circumcenter O of the antipedal triangle of G. This line does not have a name.
	www.maa.org /editorial/knot/sixcircum.shtml (1575 words)

	Triangle Special Segments
		The circumcenter is equidistant from all three vertices of the triangle.
		The circumcenter is the center of the circumscribed circle, which can be constructed through all three vertices.
		In a right triangle, the circumcenter is also the midpoint of the hypotenuse.
	mrwyatt.com /GSP/GSP_Triangle_Special_Segments.htm (467 words)

	Circumce.gsp (Site not responding. Last check: 2007-10-08)
		The circumcenter is the intersection of the perpendicular bisectors of a triangle.
		The circumcenter allows you to circumscribe a circle around a triangle.
		with the radius the length from the circumcenter to on of the verticies of the triangle.
	www.sauguscenturions.com /keaton/Circumce.htm (134 words)

	Circumcenter (Site not responding. Last check: 2007-10-08)
		Given the triangle t = ABC, its circumcenter is the intersection point O of the medial lines OJ, OD, OF of its sides BC, CA and AB respectively.
		2) The circumcircle c of q passes through the circumcenter O of the triangle t.
		For an application of these remarks in a case of determination of the focus of a parabola, look at the file ParabolaSkew.html.
	www.math.uoc.gr /~pamfilos/eGallery/problems/Circumcenter.html (295 words)

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