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Topic: Concurrent lines

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	Triangle - Wikipedia, the free encyclopedia
		Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent.
		A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e.
		Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).
	en.wikipedia.org /wiki/Triangle (2172 words)

	Chapter Two
		An incenter of a triangle is the point of concurrency of the angle bisectors of the triangle.
		An altitude of a triangle is a perpendicular segment from a vertex to the line containing the side opposite that vertex.
		The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
	pages.cthome.net /jfleary/2gnf4/2gn4c5/2gn4(5-3).htm (280 words)

	Concurrent Pipelines ABAS
		Concurrent pipelines are needed when multiple streams are co-located on a single processor and where there are real-time requirements associated with the production of final outputs.
		The purpose of the Concurrent Pipelines ABAS is to provide a reasoning framework so that the performance consequences of architectural decisions are understood at design time.
		For the Concurrent Pipelines ABAS, we consider a single processor on which multiple processes reside and are organized into sequences.
	www.sei.cmu.edu /ata/pipeline_abas.html (1159 words)

	Xah: Introduction to Real Projective Plane
		Four lines p,q,r,s, of which no three are concurrent, are the sides of a complete quadrilateral pqrs, of which the six vertices are the points qr, ps, rp, qs, pq, rs.
		Four concurrent lines a,b,c,d, are said to form a harmonic set if there is a quadrilateral of which two opoosite vertices lies on a and two other opposite vertices on b, while the remaining vertices lie on c and d.
		The points on a line are said to form a range, especially when we regard them as the possible positions of a variable point X (which runs along the line).
	xahlee.org /projective_geometry/projective_geometry.html (6405 words)

	(WO/2000/014513) SPECKLE MITIGATION FOR COHERENT DETECTION EMPLOYING A WIDE BAND SIGNAL (Site not responding. Last check: 2007-10-10)
		The concurrent independent measurements are obtained by use of multiple receiving apertures and/or by transmission of a laser signal having multiple carrier spectral lines so far apart that each carrier component of the laser signal permits performance of a measurement independent of measurements performed by laser signals at other carrier frequencies.
		This provides for the pulse train a set of sinusoidal components at line frequencies wherein the sinusoidal components are synchronized, as distinguished from randomly occurring, and the spacing of the spectral lines is large enough to provide for a decorrelation of the speckled patterns produced by each of the sinusoidal components.
		The vertical line of the image plane 40 also serves to measure distance off of boresight of the telescope 30, and the horizontal line, axis 42, also serves as a measure of intensity of radiation received at the image plane 40.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=00/14513.000803&ELEMENT_SET=DECL (6074 words)

	mathworld
		Straight lines incident with a plane (coplanar lines) and passing through a common point are said to be concurrent lines and the set of all such concurrent coplanar lines is called the pencil.
		The double ratio of four lines of a pencil is equal to the corresponding double ratio of four points which are sections of the above mentioned four lines with a transversal not incident with the base of the pencil.
		Straight lines that form the congruence or the complex are called the lines or rays of the congruence or the complex.
	ca.geocities.com /ingsaler6/mathworld.html (2193 words)

	Pappus's hexagon theorem - Wikipedia, the free encyclopedia
		Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points x, y, z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear.
		The dual of this theorem states that given one set of concurrent lines A, B, C, and another set of concurrent lines a, b, c, then the lines x, y, z defined by pairs of points resulting from pairs of intersections A∩b and a∩B, A∩c and a∩C, B∩c and b∩C are concurrent.
		A generalization of this theorem is Pascal's theorem, which was discovered by Blaise Pascal at the age of 16.
	en.wikipedia.org /wiki/Pappus's_hexagon_theorem (379 words)

	Duality
		The diagonal lines of the quadrilateral are EG, FH, and IJ.
		We assert that the diagonal lines are not concurrent.
		Suppose the diagonal lines EG, FH, and IJ are concurrent.
	www.mnstate.edu /peil/geometry/C4ProjectiveGeometry/6Duality.htm (930 words)

	Stylized demonstration of the dual of Pappus' theorem
		Indeed, that dual theorem asserts that for two triplets of concurrent lines, the joints of some of their intersections concur.
		So indeed in the applet we have two triplets of concurrent lines and its easy to verify that the joints are the same in both theorems.
		Furthermore, in terms of projective geometry the "general" theorem and the one illustrated by the applet are equivalent, for it's possible to projectively transform a "general" configuration into the one of two sets of parallel lines with the lines from two different triplets perpendicular.
	www.cut-the-knot.org /Curriculum/Geometry/PappusDual.shtml (423 words)

	Pencil of lines (Site not responding. Last check: 2007-10-10)
		, that is, a set of lines passing through the same point, is a one-dimensional projective space called a pencil of lines.
		That the space is one-dimensional is the obvious result of applying the principle of duality: a set of concurrent lines is the same as a set of collinear points.
		We will say no more about a pencil of lines other than to mention that it exists and that it has several applications in computer vision.
	vision.stanford.edu /~birch/projective/node9.html (77 words)

	Transformation of coordinates (Projective; Affine; Metric)
		It is known as the theorem of CEVA for concurrent lines.
		If the lines defined by the three pairs of corresponding vertices of two triangles are concurrent, then the intersection points of the three pairs of corresponding sides of the triangles are collinear.
		Prove that the median line from A, BC' and B'C are concurrent.
	www.ping.be /~ping1339/coortf.htm (1901 words)

	OAB - Online Information article about OAB
		original figures corresponds of course a parallel line in the other; moreover, it is seen that concurrent lines in either figure correspond to lines forming a closed polygon in the other.
		An exception to the general statement occurs when the six lines are such that they are possible lines of action of a system of six forces in equilibrium; they are then said to be in involution.
		The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant.
	encyclopedia.jrank.org /NUM_ORC/OAB.html (13797 words)

	CONCURRENT ENGINEERING
		Of all these, we have chosen that of Concurrent Engineering as this is the one that evidences most clearly the collective effort, the cooperation involved between all the agents intervening in the process.
		Concurrent Engineering is a new approach, in full development, incorporating a wide variety of new concepts and project management methods.
		The goals Concurrent Engineering attempts to meet are: a reduction of the time-to-market span, adaptation of the product to the needs or preferences of the users, efficient and low-cost maintenance, a given level of quality, all of that at a reasonable cost.
	www.iai.csic.es /netcim/concur.htm (1267 words)

	Ceva's theorem - Wikipedia, the free encyclopedia
		Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC
		Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC
		Given a triangle ABC, and points D, E, and F that lie on lines BC, CA, and AB respectively, the theorem states that lines AD, BE and CF are concurrent if and only if
	en.wikipedia.org /wiki/Ceva's_theorem (252 words)

	THE CHARM OF GEOMETRY (Site not responding. Last check: 2007-10-10)
		Prove that perpendicular lines from A’, B’, and C’ on the opposite sides of the triangle BC, CA and AB are concurrent.
		Prove that taking a line that pass through O and intersect two curves one being the inversion of the other one, than the angles between this line and the tangent lines on the intersection points with these curves are equal.
		Show that the line joining a vertex of the triangle with the center of the opposite square is equal to and perpendicular on the line that joins the centers of the other two squares.
	www.austega.com /florin/CharmOfGeometry.htm (12419 words)

	Theorem of Desargues: Let lines (AA’), (BB’) and (CC’) be three distinct and concurrent lines with D as the ...
		Theorem of Desargues: Let lines (AA’), (BB’) and (CC’) be three distinct and concurrent lines with D as the common intersecti
		Let lines (AA’), (BB’) and (CC’) be three distinct and concurrent lines with D as the common intersection point.
		the three intersection points of pairs of corresponding lines as indicated (assuming none are parallel lines).
	archive.ncsa.uiuc.edu /Classes/MATH198/whubbard/303/project1/desargues (56 words)

	PlanetMath: fundamental theorem on isogonal lines
		This theorem is a direct consequence of Ceva's theorem (trigonometric version).
		"fundamental theorem on isogonal lines" is owned by drini.
		This is version 1 of fundamental theorem on isogonal lines, born on 2002-09-02.
	planetmath.org /encyclopedia/FundamentalTheoremOnIsogonalLines.html (83 words)

	Budget Development System -- Staff Percentages
		The percent on the funding line is the percent of the employee's total pay this line is paying.
		If there are other pay lines, however, which contribute to that regular pay for that same time period, then the percent for ALL the pay lines for that same time period must add up to no more than 100%.
		Using a generic account, rather than leaving out the pay line altogether, will help you calculate the percent time/effort on all of the pay lines correctly, and make sure that you make the person " whole" when figuring out all of the sources of their pay.
	www.usc.edu /dept/finserv/dirtrng/bds-staff.htm (1507 words)

	Cut The Knot!
		Divide an angle into n equal parts with (n-1) lines, remove all the lines but the extreme two - the ones which are next to the sides of a given triangle.
		Toying with the applet provides a convincing demonstration that in all three families the lines are indeed concurrent regardless of the value of n.
		In that framework, concurrency of the lines PU, QV, RW is deduced from a result in Analytic Geometry illustrated by the following applet.
	www.maa.org /editorial/knot/morley.html (1757 words)

	Projective Geometry
		We have a set of points known as a plane with special subsets known as lines, sets which are unions of points and are denoted as lower-case italic letters.
		We have a set of lines known as a plane with special subsets known as points, sets which are intersections of lines and are denoted as lower-case italic letters.
		We call the set of all lines containing a given point a pencil and the set of all points lying on a given line a range.
	halogen.note.amherst.edu /~wing/project/content.php?page=2 (618 words)

	think again! - comments
		Kobon Fujimura learnt in school that three straight lines are needed to draw a triangle.
		For four lines he found two triangles was the most he could produce.
		Two of the triangles formed by the 5 lines, each of which is cut into two triangles by the sixth line, are not counted in the set of triangles formed by the 6 lines.
	simpler-solutions.net /pmachinefree/thinkagain/comments.php?id=558_0_3_0_C (192 words)

	BBC - Education Scotland - Higher Bitesize Revision - Maths - Geometry - The straight line: Revision 6
		Three or more lines are said to be concurrent if they all pass through a common point.
		are concurrent and state the point of concurrency.
		The first two lines are as in the previous example.
	www.bbc.co.uk /scotland/education/bitesize/higher/maths/geometry/the_straight_line6_rev.shtml (195 words)

	Streamed Lines: Branching Patterns for Parallel Software Development
		Streamed Lines is a pattern language that attempts to provide at least a partial answer to this question by presenting branching and merging patterns for decomposing a project's workflow into separate lines of development, and then later recomposing these lines back into the main workstream.
		In general, when a branch corresponds to a line of development containing (or intended for) multiple sets of logical changes, we refer to the branch as a codeline, even though it need not be limited to source-code artifacts.
		Branches and codelines are indicated with solid lines, whereas merges and propagations are indicated with dashed lines.
	www.cmcrossroads.com /bradapp/acme/branching (6766 words)

	MathLinks Math Forum :: View topic - n lines, n-2 triangles (Site not responding. Last check: 2007-10-10)
		We are given n lines in the plane, in general postion, no 2 paralel, no 3 concurent.
		Lines are moved parralel to their initial postions.
		The idea is to move lines, but to keep the sizes of the triangles.
	www.mathlinks.ro /Forum/topic-6246.html (1067 words)

	Assignment2.html
		That is, for three lines to be concurrent, there must be one and only one point of intersection among the three lines.
		Since line AB is not parallel to line BC, it follows that the perpendicular bisector of side AB, namely line k, cannot be parallel to the perpendicular bisector of side BC, namely line m.
		Since any point on line n is equidistant from A and C, and since P is equidistant from A and C, it follows that P is on line n.
	web.pdx.edu /~dhh/assignment2.html (725 words)

	UNIT (Site not responding. Last check: 2007-10-10)
		intersection as Z. Draw a line through point P and Z. PZ is perpendicular to l at P. Given a point outside the line, construct the perpendicular to the line* from the given point.
		The students will define concurrent in their notes after examples of concurrent lines have been given by the instructor.
		They will then discuss the definition of concurrent lines and will be asked to write this definition in their notes along with a picture of their own interpretation.
	www.usd.edu /~vschultz/Unit2.htm (3450 words)

	hyperbolic page
		In such cases, care may be needed to describe conditions on the i-lines.
		Since the concurrences are not at p, they are at the other intersection of C(b,r,p),C(c,q,p).
		The i-line C(p,q,r) is called the simpson z-line for the z-triangle ABC for the point m.
	www.maths.gla.ac.uk /~wws/cabripages/misc/misc1.html (603 words)

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