21. Sci.math FAQ: The Trisection Of An Angle Vorherige Nächste Index sci.math FAQ The trisection of an angle. Figure7.1 Trisection of the Angle with a marked ruler Let theta be angle BAO. http://www.uni-giessen.de/faq/archiv/sci-math-faq.trisection/msg00000.html

Index sci.math FAQ: The Trisection of an Angle Subject : sci.math FAQ: The Trisection of an Angle From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : Fri, 27 Feb 1998 19:38:59 GMT Newsgroups sci.math news.answers sci.answers Sender news@undergrad.math.uwaterloo.ca (news spool owner) Summary : Part 18 of 31, New version http://daisy.uwaterloo.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Index: Index sci-math-faq.trisection

22. Trisecting An Angle Response 3 of 3 Author jlu Here is an intuitive argument that may make itclearer why trisection of an angle is impossible with ruler and compass. http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH101.HTM

Ask A Scientist Mathematics Archive Trisecting an angle Back to Mathematics Ask A Scientist Index NEWTON Homepage Ask A Question ... NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.

23. Trisection Of An Angle - Encyclopedia Article About Trisection Of An Angle. Free encyclopedia.thefreedictionary.com/Angle%20trisection trisection . A highly accurate approximate constructionby Mark Stark. Drag the point B to change the angle AOB Drag the http://encyclopedia.thefreedictionary.com/Trisection of an angle

Dictionaries: General Computing Medical Legal Encyclopedia Trisection of an angle Word: Word Starts with Ends with Definition A number of ancient problems in geometry Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible of proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. Click the link for more information. involve the construction of lengths or angles This article is about angles in geometry. See: Fly fishing for the technique of using a bait and hook to catch fish. Angles for the Germanic tribe that moved to Britain. Angle, Pembrokeshire for the place in Wales. Angle (professional wrestling) for 'angles' in professional wrestling. An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles are studied in geometry and trigonometry. Click the link for more information.

24. Trisecting The Angle trisection of an angle. Given angle ABC, trisect angle ABC. Step1. This requires the use of a marked straight edge. This can be http://www.geocities.com/robinhuiscool/Trisectionofangle.html

TRISECTION OF AN A N G L E Given angle ABC, trisect angle ABC. Step 1. This requires the use of a marked straight edge. This can be something like a slip of paper or ruler. First draw a line parallel to line BC at point A. Step 2. Draw a perpendicular line from point A, intersecting BC at D. Step 3. Mark off on the straight edge points E,F and G where EF=FG=AB. Step 4. Position the straight edge so that it crosses point B, point E touches AD, and point G touches line A. Angle CBG is 1/3 of ABC.

25. Haitian Math Whiz May Have Unraveled Age ramifications are huge. The trisection of an angle is one of the infamous threeproblems of antiquity which have stumped mathematicians for centuries. http://www.radiolakay.com/romain.htm

please download macromedia Listen Live : Real Player or Windows Media Player Live Chat with Radio Lakay on AIM chat AIM:LAKAYINTER Links Global Outlook Library of Congress Bob Corbett's Home Page Haitian Math Whiz May Have Unraveled Age-old Geometry Mystery HAITI PROGRES ( http://www.haiti-progres.com ), October 9 - 15, 2002 Vol. 20, No. 30 by Kim Ives PHOTO: Leon Romain has devised a theorem for trisecting any angle, one of geometry's great puzzles. If he is right, it could change your life. So far, nobody has proved him wrong. Around 450 B.C., the Greek mathematician, Hippias of Ellis, began searching for a way to trisect an angle. Over 2000 years later, in 1837, a French mathematician named Pierre Wantzel proclaimed that it was impossible to trisect an angle using just a compass and a straightedge, the only tools allowed in geometric construction. But now, at the dawn of the twenty-first century, a Haitian computer program designer, Leon Romain, claims he has proven, with a "missing theorem," that it is possible to trisect an angle with those simple tools, disproving Wantzel's assertion and exploding centuries of mathematical gospel. "This discovery shows us that the notions that every mathematician has held for the past 200 years as absolute

26. Akolad News| Romain The trisection of an angle is one of the infamous three problemsof antiquity which have stumped mathematicians for centuries. http://www.akolad.com/news/romain.htm

Haitian Math Whiz May Have Unraveled Age-old Geometry Mystery HAITI PROGRES ( http://www.haiti-progres.com), October 9 - 15, 2002 Vol. 20, No. 30 by Kim Ives PHOTO: : Leon Romain has devised a theorem for trisecting any angle, one of geometry's great puzzles. If he is right, it could change your life. So far, nobody has proved him wrong Around 450 B.C., the Greek mathematician, Hippias of Ellis, began searching for a way to trisect an angle. Over 2000 years later, in 1837, a French mathematician named Pierre Wantzel proclaimed that it was impossible to trisect an angle using just a compass and a straightedge, the only tools allowed in geometric construction. But now, at the dawn of the twenty-first century, a Haitian computer program designer, Leon Romain, claims he has proven, with a "missing theorem," that it is possible to trisect an angle with those simple tools, disproving Wantzel's assertion and exploding centuries of mathematical gospel. "This discovery shows us that the notions that every mathematician has held for the past 200 years as absolute certainty are actually false," Romain told Haiti Progres. "The mathematical and even philosophical ramifications are huge."

27. Newsgroups Sci.math From Ramsay@unixg.ubc.ca (Keith Ramsay) Newsgroups sci.math From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz) SubjectRe NEW sci.math FAQ The trisection of an angle Date Thu, 10 Nov 1994 01 http://www.math.niu.edu/~rusin/known-math/93_back/trisect

Newsgroups: sci.math From: ramsay@unixg.ubc.ca (Keith Ramsay) Subject: Re: Trisect an angle? Date: Thu, 17 Sep 1992 17:26:11 GMT In article jurjus@kub.nl (H. Jurjus) writes: > In Article Greg Griffiths

28. Math Forum: Ask Dr. Math: FAQ trisection of an angle Given an angle, construct an angle one third aslarge. (Some angles, such as 90 degrees, can be trisected easily.). http://www2.sunysuffolk.edu/wrightj/MA28/Greek/Impossible.htm

Ask Dr. Math: FAQ "I mpossible" G eometric C onstructions Three geometric construction problems from antiquity puzzled mathematicians for centuries: the trisection of an angle, squaring the circle, and duplicating the cube. Are these constructions impossible? Whether these problems are possible or impossible depends on the construction "rules" you follow. In the time of Euclid, the rules for constructing these and other geometric figures allowed the use of only an unmarked straightedge and a collapsible compass. No markings for measuring were permitted on the straightedge (ruler), and the compass could not hold a setting, so it had to be thought of as collapsing when it was not in the process of actually drawing a part of a circle. Following these rules, the first two problems were proved impossible by Wantzel in 1837, although their impossibility was already known to Gauss around 1800. The third problem was proved to be impossible by Lindemann in 1882. The impossibility proofs depend on the fact that the only quantities you can obtain by doing straightedge-and-compass constructions are those you can get from the given quantities by using addition, subtraction, multiplication, division, and by taking square roots. These numbers are called Euclidean numbers, and you can think of them as the numbers that can be obtained by repeatedly solving the quadratic equation. These three problems require either taking a cube root or constructing pi. A cube root is not a Euclidean number, and Lindemann showed that pi is a transcendental number, which means that it is not the root of an algebraic equation with integer coefficients, making it too non-Euclidean.

29. The Geometer's Sketchpad® - JMM Activities The Conchoid and Trisection of Angles. It can also be used to trisect angles.An illustration of the curve and the trisection of an angle will be given. http://www.keypress.com/sketchpad/usergroups/jmm2004/

Home Resources Technical Support JavaSketchpad ... Site Map Resources Bibliography 101 Project Ideas Sketchpad Links Sketchpad for Cassiopeia ... TI Graphing Calculators JavaSketchpad About JSP JSP Gallery JSP Download Center JSP Developer's Grammar ... JSP Links Instructor Resources Download Instructor's Evaluation Edition Workshop Guide Professional Development Recent Talks Technical Support FAQ Product Updates Tech Support Request Form General Information Product Info How to Order Curriculum Modules Other Key Sites Key Curriculum Press Key College Publishing KCP Technologies Keymath.com Activities from the Joint Mathematics Meeting Phoenix, Arizona January 2004 The Sketchpad User Group had its first meeting of 2004 at the Joint Meetings of the American Mathematical Society and the Mathematical Association of America in Phoenix. This page collects the activities and sketches that several participants shared with their colleagues at the User Group meeting. You can download these Word files and Sketchpad documents individually, or as a complete collection in a .zip archive

30. Sci.math FAQ: The Trisection Of An Angle Subject sci.math FAQ The trisection of an angle. Figure 7.1 Trisectionof the Angle with a marked ruler Let theta be angle BAO. http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/trisection.html

Note from archivist@cs.uu.nl : This page is part of a big collection of Usenet postings, archived here for your convenience. For matters concerning the content of this page , please contact its author(s); use the source , if all else fails. For matters concerning the archive as a whole, please refer to the archive description or contact the archivist. Subject: sci.math FAQ: The Trisection of an Angle This article was archived around: Fri, 27 Feb 1998 19:38:59 GMT All FAQs in Directory: sci-math-faq All FAQs posted in: sci.math Source: Usenet Version Archive-name: sci-math-faq/trisection Last-modified: February 20, 1998 Version: 7.5 http://daisy.uwaterloo.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick

31. M3210 Sample Exam II trisection of an angle The problem is to find the angle trisectorsfor an arbitrary angle. The general problem can not be done http://www-math.cudenver.edu/~wcherowi/courses/m3210/hgex2sam.html

Sample Exam II . Define the following italicized terms by completing the sentances below. (Your definitions need not be verbatim reproductions of the book or class notes, but they must be correct!) a) The incenter of a triangle is ... the point of intersection of the internal angle bisectors of the triangle (or the center of the inscribed circle). b) The Euler line of a triangle is the line segment determined by ... the circumcenter (O) and the orthocenter (H) of the triangle. c) The centroid of a triangle is ... the point of intersection of the medians of the triangle. d) A constructible number is the length of a line segment which ... can be obtained from a unit length by straightedge and compass constructions. e) The 9-point circle of a triangle contains the following 9 points of a triangle : the three midpoints of the sides of the triangle, the three feet of the altitudes of the triangle and the three midpoints of the segments drawn from the vertices to the orthocenter of the triangle. Prove that the internal bisectors of two angles of a triangle and the external bisector of the third angle intersect the opposite sides of the triangle in three collinear points. Solution Using compass and straightedge construct the orthocenter of the triangle below.

32. Demonstration Of The Archimedes' Solution To The Trisection Problem analog device simulation for drawing ellipses The problem of constructing an angle equal to the one third of the given one has been pondered The solution for the angle trisection can be presented in a more http://www.cut-the-knot.com/pythagoras/archi.html

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Angle Trisection by Archimedes of Syracuse (circa 287 - 212 B.C.) Archimedes of Syracuse is popularily known for the law he discovered on occasion of taking his bath . "Eurika" he exclaimed and made it into the history. (Along with Newton and Gauss he is counted among the greatest mathematicians of all times. As an engineer he frustrated numerous attempts by the Romans to capture the city of Syracuse.) The problem of constructing an angle equal to the one third of the given one has been pondered since the times of antiquity. Probably to make the notion of 'geometric construction' more exciting the Ancient Greeks have restricted the allowed operations to using a straightedge and a compass. It's thus specifically forbidden to use a ruler for the sake of measurement. Three famous construction problems lingered until early 19th century when it was shown that it's impossible to solve them with the help of only a straightedge and a compass. The three problems are: to trisect a given angle, to double a cube, and to square a circle . However, one illicit solution that has been found in the works of Archimedes is demonstrated below.

33. Angle Trisection -- From MathWorld angle trisection. angle trisection is the division of an arbitrary angle into threeequal angles. Ogilvy, C. S. angle trisection. Excursions in Geometry. http://mathworld.wolfram.com/AngleTrisection.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Geometric Construction Geometry ... Angles Angle Trisection Angle trisection is the division of an arbitrary angle into three equal angles . It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836). Although trisection is not possible for a general angle using a Greek construction, there are some specific angles, such as and can be trisected. Furthermore, some angles are geometrically trisectable, but cannot be constructed in the first place, such as (Honsberger 1991). In addition, trisection of an arbitrary angle can be accomplished using a marked ruler (a Neusis construction ) as illustrated above (Courant and Robbins 1996). An angle can also be divided into three (or any whole number ) of equal parts using the quadratrix of hippias or trisectrix An approximate trisection is described by Steinhaus (Wazewski 1945, Steinhaus 1999, p. 7). Given an angle

34. Angle Trisection angle trisection. Most people are familiar from high school geometry with compass and straightedge constructions. For instance I remember being taught how to bisect an angle, inscribe a square into a http://www.geom.umn.edu/docs/forum/angtri

Up: Geometry Forum Articles Angle Trisection Most people are familiar from high school geometry with compass and straightedge constructions. For instance I remember being taught how to bisect an angle, inscribe a square into a circle among other constructions. A few weeks ago I explained my job to a group of professors visiting the Geometry Center . I mentioned that I wrote articles on a newsgroup about geometry and that sometimes people write to me with geometry questions. For instance one person wrote asking whether it was possible to divide a line segment into any ratio, and also whether it was possible to trisect an angle. In response to the first question I explained how to find two-thirds of a line segment. I answered the second question by saying it was impossible to trisect an angle with a straightedge and a compass, and gave the person a reference to some modern algebra books as well as an article Evelyn Sander wrote about squaring the circle . One professor I told this story to replied by saying, "Bob it is possible to trisect an angle." Before I was able to respond to this shocking statement he added, "You just needed to use a MARKED straightedge and a compass." The professor was referring to Archimedes' construction for trisecting an angle with a marked straightedge and compass. When someone mentions angle trisection I immediately think of trying to trisect an angle via a compass and straightedge. Because this is impossible I rule out any serious discussion of the manner. Maybe I'm the only one with this flaw in thinking, but I believe many mathematicians make this same serious mistake.

35. Trisecting An Angle Now this is exactly the curve needed to solve both versions of trisection ofan angle given above and Nicomedes solved the problem with his curve. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trisecting_an_angle.html

Trisecting an angle Ancient Greek index History Topics Index There are three classical problems in Greek mathematics which were extremely influential in the development of geometry. These problems were those of squaring the circle, doubling the cube and trisecting an angle. Although these are closely linked, we choose to examine them in separate articles. The present article studies the problem of trisecting an arbitrary angle. In some sense this is the least famous of the three problems. Certainly in ancient Greek times doubling of the cube was the most famous, then in more modern times the problem of squaring the circle became the more famous, especially among amateur mathematicians. The problem of trisecting an arbitrary angle, which we examine here, is the one for which I [EFR] have been sent the largest number of false proofs during my career. It is an easy task to tell that a 'proof' one has been sent 'showing' that the trisector of an arbitrary angle can be constructed using ruler and compasses must be incorrect since no such construction is possible. Of course knowing that a proof is incorrect and finding the error in it are two different matters and often the error is subtle and hard to find. There are a number of ways in which the problem of trisecting an angle differs from the other two classical Greek problems. Firstly it has no real history relating to the way that the problem first came to be studied. Secondly it is a problem of a rather different type. One cannot square any circle, nor can one double any cube. However, it is possible to trisect certain angles. For example there is a fairly straightforward method to trisect a right angle. For given the right angle

36. An Angle Trisection an angle "trisection" A highly accurate approximate construction by Mark Stark. Drag the point B to change the angle AOB. Drag the point E to change the initial guess. The angle AOE' is a very close http://www.math.umbc.edu/~rouben/Geometry/trisect-stark.html

An angle "trisection" A highly accurate approximate construction by Mark Stark Drag the point B to change the angle AOB Drag the point E to change the initial guess The angle AOE' is a very close to being 1/3 of angle AOB Note how insensitive G and E' are with respect to displacements of E Type "r" to reset the diagram to its initial state The construction shown above, which trisects an arbitrary angle with great accuracy, was first proposed by Mark Stark in the geometry-puzzles discussion list. In a followup article Eric Bainville noted that the iteration of this trisection algorithm "will effectively converge to the trisection with a cubic convergence rate." Here is the outline of the construction, as restated by Mark in a later article Draw an arc with origin at O crossing both lines of the angle at points A and B. Draw line AB making an isosceles triangle. Using point A as the origin, draw an arc crossing line AB and the earlier arc somewhere between 1/4 and 1/2 way between points A and B. Label where this new arc crosses line AB point D. Label where this new arc crosses the first arc point E. Draw line DE and extend it well past O. If line DE passes exactly through point O (it wont) stop, your first guess was an exact trisection.

37. Angle Trisection angle trisection. When someone mentions angle trisection I immediatelythink of trying to trisect an angle via a compass and straightedge. http://www.geom.uiuc.edu/docs/forum/angtri/

Up: Geometry Forum Articles Angle Trisection Most people are familiar from high school geometry with compass and straightedge constructions. For instance I remember being taught how to bisect an angle, inscribe a square into a circle among other constructions. A few weeks ago I explained my job to a group of professors visiting the Geometry Center . I mentioned that I wrote articles on a newsgroup about geometry and that sometimes people write to me with geometry questions. For instance one person wrote asking whether it was possible to divide a line segment into any ratio, and also whether it was possible to trisect an angle. In response to the first question I explained how to find two-thirds of a line segment. I answered the second question by saying it was impossible to trisect an angle with a straightedge and a compass, and gave the person a reference to some modern algebra books as well as an article Evelyn Sander wrote about squaring the circle . One professor I told this story to replied by saying, "Bob it is possible to trisect an angle." Before I was able to respond to this shocking statement he added, "You just needed to use a MARKED straightedge and a compass." The professor was referring to Archimedes' construction for trisecting an angle with a marked straightedge and compass. When someone mentions angle trisection I immediately think of trying to trisect an angle via a compass and straightedge. Because this is impossible I rule out any serious discussion of the manner. Maybe I'm the only one with this flaw in thinking, but I believe many mathematicians make this same serious mistake.

38. Four Problems Of Antiquity These are Doubling of the cube Construct a cube whose volume is double that ofa given one. angle trisection Trisect an arbitrary angle. angle trisection. http://www.cut-the-knot.org/arithmetic/antiquity.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Four Problems Of Antiquity Three geometric questions raised by the early Greek mathematicians attained the status of classical problems in Mathematics. These are: Doubling of the cube Construct a cube whose volume is double that of a given one. Angle trisection Trisect an arbitrary angle. Squaring a circle Construct a square whose area equals that of a given circle. Often another problem is attached to the list: Construct a regular heptagon (a polygon with 7 sides.) The problems are legendary not because they did not have solutions, or the solutions they had were unusually hard. No, numerous simple solutions have been found yet by Greek mathematicians. The problem was in that all known solutions violated an important condition for this kind of problems, one condition imposed by the Greek mathematicians themselves: Valid solutions to the construction problems are assumed to consist of a finite number of steps of only two kinds: drawing a straight line with a ruler (or rather a straightedge as no marks are allowed on the ruler) and drawing a circle. You are referred to solutions of problems and as examples of existent solutions. That no solution exists subject to the self-imposed constraints have been proven only in the 19th century.

39. Dividing One Angle Into Three Equal Angles Seems A Trivial Problem With this curve, the problem of trisecting an angle could be reduced to thetrisection of a line segment. Thus, angle CAB is the one to be trisected. http://www.perseus.tufts.edu/GreekScience/Students/Tim/Trisection.page.html

Dividing one angle into three equal angles seems a trivial problem. That is probably why it irked the Greeks so. Instead of being a simple problem, it is a complex, non-planar problem, as the Greeks soon discovered. The trisection problem can probably credit its origin to the construction of regular polygons. The discover of the construction of a perfect pentagon(see The Golden Section One of the earliest ways discovered was that of Hippias of Elis(circa 425 BC). Hippias used a curve he had invented, called the quadratrix . With this curve, the problem of trisecting an angle could be reduced to the trisection of a line segment. The following picture is one construction of such segment trisect. The great benefit of this method was that it could be generalized to divide any angle into any number of parts. I don't really like this next solution, but maybe you will. This second method, perhaps the most well known of all, can be credited to Nicomedes(circa 180 BC). Nicomedes created a special device to use in his construction. As the upper part slide back and forth in its groove, the angle of the pointer changed so as to describe a curve known as a conchoid(as a function, y=K(x^2 + C)^(-1/2) is the simplest form).

40. Explorations In Math However, attempts to use this simple trisection on an angle quickly proveduseless as you can see in Fig4. We have an angle we want trisected. http://jwilson.coe.uga.edu/emt669/Student.Folders/Godfrey.Paul/work/proj2/tri.ht

Tri as I Might by Paul Godfrey This exploration looks at various ways to trisect an angle. First we look at using an unmarked straight-edge and compass. We will also look at trisections that can be performed with marked straight-edge and compass. Then we explore using trisectrices to perform the job. Geometer's Sketchpad [1] was used for most of these explorations. For those having GSP, the GSP files can be obtained by clicking on the figure number. Reading the College Mathematics Journal [2] I noticed an article about trisecting an angle . It talked about something called a trisectrix. A trisectrix is a curve that can help us easily trisect an angle. One example given was a curve with equation called a trisectrix of Maclaurin. A graph of this curve looks like this Fig-1 next At first, this sounded like a complicated way to trisect an angle. After all, trisecting a line was a simple matter as Fig-2 shows. Fig-2 next Further, we know that given triangle DBC with rays BJ and BK as shown, any line segment parallel to DC with endpoints on rays BD and BC will be trisected by the rays BD, BJ, BK, BC due to the proportionality principle of similar triangles. Fig-3 next So, we reason that the angle we wish to trisect could be trisected using this method. Since the arc on a circle defined by the legs of the angle is the same measure as the angle, we merely need to trisect the arc. However, attempts to use this simple trisection on an angle quickly proved useless as you can see in

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