41. Explorations In Math However, attempts to use this simple trisection on an angle quicklyproved useless as you can see in Fig4. We start with angleABC. 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

42. 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

43. Angle Trisection angle trisection. Give four examples of constructive angle trisection methodsthat actually trisect an angle. Give a proof that each method works. http://www.math.tamu.edu/~dallen/m629_02a/trisection.htm

Angle trisection Give four examples of constructive angle trisection methods that actually trisect an angle. Give a proof that each method works. Show what is wrong with the method from the viewpoint of compass and straightedge constructions. Be sure to show a diagram for each. If there is a creator of the method, include the name. Prove that any angle with angle given by an integer multiple of nine can be trisected. (For example, the angles of 9, 18, 27, ... can all be trisected.) Hint. What angles can you construct at this point? What operations can you perform with angles? Add? Subtract? Etc. Cite your references - particularly Internet URL's.

44. The Regular Nine-gon And Angle Trisection angle trisection and the regular ninegon. We know that it is impossibleto trisect an angle using only a straight-edge and compass. http://www.nevada.edu/~baragar/geom/nine.html

Angle trisection and the regular nine-gon We know that it is impossible to trisect an angle using only a straight-edge and compass. Since Geometer's Sketchpad mimics such constructions, one cannot write a script or sketch that trisects an arbitrary angle (using only the buttons and construction pull down menu in sketchpad.) However, one can create a sketch that mimics Archimede's trisection using a notched straight-edge. Such a sketch is below. The one step that is not a valid construction must be done by hand. In the sketch at the right, select the (acute) angle CAB to be trisected by moving the point C . Now, move P so that the line PQ goes through C . The angle CQB is one third of the angle CAB Sorry, this page requires a Java-compatible web browser. The step that must be done by hand moving P so that PQ goes through C is the step which is not a valid construction. The regular pentagon is constructible. Thus, one can write a sketch which produces a regular pentagon inscribed in a given circle (see below). The regular nine-gon, on the other hand, cannot be constructed using only a straight-edge and compass. But, one can use Archimedes' construction. This is done below. Again, one can adjust the circle in which a regular nine-gon is to be constructed by moving A and B . Try doing this. Note how the figure is distorted. Now, adjust

45. Angle Trisection And K-Section angle trisection using just a compass and a ruler is an old problem,and of course is actually not solvable. In fact, proving so http://www.ffd2.com/recmath/ksect.html

Angle trisection using just a compass and a ruler is an old problem, and of course is actually not solvable. In fact, proving so is apparently often assigned in some pure math courses. Angle trisection amounts to finding the roots of a certain cubic equation, which cannot be done except in special cases (trisecting 270 degrees is just not that tough). Waaaay back in ninth grade, in Mr. Laeser's geometry class, I brought up angle trisection, knowing next to nothing about it. So, Mr. Laeser said he'd give extra credit to anyone who could trisect an angle using a compass and a straightedge. My attempts were dismal failures, and I think the only other person who attempted it was my friend George (of the famous GASUC) Wright, who spent some more time with it (I think he got nine red squares for his efforts). Anyways, the summer of 1993 I was eating lunch with Bill Beyer, one of the Old Bulls at Los Alamos, and he suggested that I derive a method for angle trisection. Well, when the gauntlet is dropped like that there is simply no choice in the matter. So I sat down that afternoon and thought about it and by lunchtime the next day I had my trisection method all worked out. So then he said that what he really wanted was a method of n-secting an angle, so that afternoon I sat down and figured that one out (it is really a very simple extension of the trisection algorithm, so no big deal). After this I did a little research into trisection methods. I don't remember much of what I found, except that most of the methods are clever geometric methods that are an awful lot easier to implement than my method. I wrote up my stuff in some little .tex papers: the trisection trilogy. I never submitted them to anyone, but I put them here in their original form, warts and all. This was done long before I knew about things like iterative maps, and there are all sorts of little errors, Way Too Much Information sections, foolish statements, etc.

46. Perseus Update In Progress quadratrix. With this curve, the problem of trisecting an angle couldbe reduced to the trisection of a line segment. The following http://www.perseus.tufts.edu/GreekScience/Students/Tim/Trisection.page.html

The Perseus Digital Library is Being Updated Notice The main Perseus web site (at Tufts) is unavailable from 5:00 to 6:00, US Eastern time, in order to rebuild its databases with new or changed meta-data. We apologize for this inconvenience.

47. Trisection On A Budget trisection On A Budget. It's well known that there is no procedure using just straightedgeand compass for trisecting an arbitrary angle in a finite number of http://www.mathpages.com/home/kmath169.htm

Trisection On A Budget It's well known that there is no procedure using just straight-edge and compass for trisecting an arbitrary angle in a finite number of steps. However, we certainly can trisect an arbitrary angle to within any arbitrary precision by means of very simple straight-edge and compass operations. One approach is to simply bisect the angle, then bisect the left half-angle, then the right quarter-angle, then the left eighth-angle, and so on. The net result is 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - ... which differs from 1/3 by less than 1/2^n after n bisections. This may not be very elegant, but it's easy to remember and gives a construction with as much (finite) precision as you want. With 30 bisections the result would be within 1 part in 10^9 of a true trisection. "Hope deceives more men than cunning does." Vauvenargues, 1746 Return to MathPages Main Menu

48. Demonstration Of The Archimedes' Solution To The Trisection Problem Analog device simulation for drawing ellipses angle trisection. by Archimedes of Syracuse. (circa 287 212 B.C.) 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.

49. A Real Trisection program). It interactively demonstrates the actual trisection of anangle. Move the bright red dot in order to change the angle. http://www.jimloy.com/cindy/trisect1.htm

Return to my Cinderella pages Return to my Mathematics pages Go to my home page A Real Trisection The following is an interactive Java applet, created with Cinderella (a geometry program). It interactively demonstrates the actual trisection of an angle. Move the bright red dot in order to change the angle. That dot is a hinge, as are the other red dots that far from the vertex. The two red dots farthest from the vertex are hinges for the short bars, but slide along the angle trisectors. The whole device is just a couple of parallelograms and a couple diagonals. Please enable Java for an interactive construction (with Cinderella). We cannot use a pair of compasses and a straightedge to trisect a general angle. That has been proven. But we can make other tools (such as the device simulated here) to trisect such an angle. In fact, we can use a pair of compasses and a straightedge to make this device. All that this device does is triple the smallest angle. Return to my Cinderella pages Return to my Mathematics pages Go to my home page

50. Demonstration Of The Archimedes' Solution To The Trisection Problem angle trisection by Archimedes of Syracuse (circa 287 212 BC) Hi Alex. The solutionfor the angle trisection can be presented in a more straightforward way. http://www.cut-the-knot.com/pythagoras/archi.shtml

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.

51. Trisection De L'angle trisection DE L'angle. ÉQUATION. Trisectingthe angle by Steven Dutch. angle trisection The Geometry Center. http://membres.lycos.fr/villemingerard/Histoire/Trisangl.htm

Accueil Dictionnaire Rubriques Index ... M'écrire Édition du: Rubrique: Histoire antiquité Introduction Duplication du cube Trisection de l'angle Quadrature du cercle ... Heptagone Sommaire de cette page ÉQUATION Pages voisines Règle et compas Transcendant Histoire Hilbert ... Bissection Trisection Découper un angle quelconque en deux parts égales Découper un angle quelconque en trois parts égales Bissection Découper un angle quelconque en deux parts égales est facile Pourquoi est-ce si difficile pour trois? ÉQUATION Idée de la démonstration avec un angle de 20° Calculons en général cos(3a) = cos(a)cos(2a) - sin(a)sin(2a) = cos(a)(cos (a) - sin (a)) - 2sin (a)cos(a) = cos(a)(2cos (a) - 1) - 2(1 - cos (a))cos(a) (a) - 3cos(a) Prenons le cas particulier de a o cos(3a) = cos(60 o L'équation, dans ce cas, devient (a) - 3cos(a) (a) - 6cos(a) - 1 En remplaçant cos(a) = x Avec v = 2x = v Voir Équation Solutions rationnelles ? Supposons que Oui, alors v = p/q fraction minimale (simplifiée) En remplaçant dans l'équation = (p/q) - 3(p/q) - 1 En multipliant par q = p - q En reformulant q = p = p (p² - 3q²) On déduit que p est divisible par q Conséquence p est divisible par q Impossible p/q est une fraction irréductible par hypothèse Et en factorisant avec p p + q = q (3p + q²) On déduit que q est divisible par p Conséquence q est divisible par p Impossible p/q est une fraction irréductible par hypothèse La supposition est fausse v n'est par rationnel En généralisant Il n'est pas possible diviser un angle par construction Démonstration en 1837 par Pierre Laurent

52. Trisection Selon Nicomède Translate this page trisection de l'angle selon Nicomède. Nicomède proposa une solutionapprochée de la trisection de l'angle par la construction http://www.sciences-en-ligne.com/momo/chronomath/anx3/trisection.html

Trisection de l'angle selon prop trisection de l'angle par la construction d'une trisectrice l'angle ^AOB = ^OJA = 2t, c'est dire que ^xOA = 3t. Vous voyez se construire point par point la branche ( G ) de la trisectrice G trisection de l'angle ^xOA. r = OP + PP' = OK/cos t + 2a avec t p C'est une (prononcer ) de la droite (AK). G r = 1/cos t + 2 p . Les portions ( G G 2) et ( G p p p p p Maclaurin Thomas Ceva Morley : Pour en savoir plus Ed. Hermann, Paris - 1989

53. Hippias D'Elis Translate this page Cherchant à résoudre le problème de la trisection de l'angle , il inventa unecourbe trisectrice permettant une solution approchée (construction point par http://www.sciences-en-ligne.com/momo/chronomath/chrono1/Hippias.html

HIPPIAS d'Elis grec, vers -450 Philosophe sophiste, diplomate, il connut Socrate trisection de l'angle , il inventa une courbe trisectrice quadratrice de Dinostrate car ce dernier l'utilisa pour tenter la quadrature du cercle. Wantzel en 1837 : Exercice : Exercices : Etude de l'ennéagone régulier (trisection d'un angle de 60°) Trisection d'une diagonale Trisectrice/quadratrice de Dinostrate : Trisectrice selon Nicomède : Hippocrate de Chio

54. La Trisection De L'angle Translate this page Positionner le point S jusqu'à ce que le côté droit de l'angle rosesoit tangent au cercle. Alors la trisection apparaitra aussitôt. http://perso.wanadoo.fr/therese.eveilleau/pages/truc_mat/textes/trisection.htm

L e trisecteur et les trisectrices... Manipulons L'instrument Exp l ications On pourrait se faire angle Et, sinon vivre au calme, Attaquer l'entourage, Se reposer ensuite Guillevic 1967 M anipulons L e trisecteur F A L' P ositionner le point S L 'instrument M ode d'emploi : E xplic ations N AI = IJ = JB et l'angle AIS est droit. Mesure d'audience et statistiques Classement des meilleurs sites, chat, sondage

55. Trisection D'un Angle Translate this page La trisection d'un angle par pliage 1) Tracer une droite jaune parallèle à (AB)2) D est le symétrique du point A 3) Il faut trouver la droite rouge axe du http://perso.wanadoo.fr/math.lemur/hub2d/anglepli.htm

La trisection d'un angle par pliage 1) Tracer une droite jaune parallèle à (AB) 2) D est le symétrique du point A 3) Il faut trouver la droite rouge axe du pliage qui envoie D sur la droite (AC) et A sur la droite jaune D'où connaissance des points K et J La trisectrice de Maclaurin Tracer une droite passant par A parallèle à la droite (OM) Tracer une droite passant par A parallèle à la droite (OM')

56. Euclid Challenge - Trisection Of Any Angle By Straightedge And Compass - Page 4 May 10, 2002. Page 4 trisection of Any angle by Straightedge and Compass.Note 1 3. 45º 90º, Bisect angle, 22½º - 45º, trisection X 2. http://www.euclidchallenge.org/pg_04.htm

EUCLID CHALLENGE Successful Response by Milton Mintz May 10, 2002 Page 4: Trisection of Any Angle by Straightedge and Compass Note 1: Basic range of angles Adjustment before trisection Range of angles after adjustment Adjustment after trisection Between: Between: Add 22 Trisection minus 7 None None Bisect angle Trisection X 2 Take ¼ of angle Trisection X 4 Note 2: EXAMPLE ANGLE: 30 Since 60º is a frequent test angle, the above 30º example was used so that the resulting trisection could be doubled. Previous Page Top of Page Next Page

57. Euclid Challenge - Trisection Of Any Angle By Straightedge And Compass - Page 9 Successful Response by Milton Mintz. May 10, 2002. Page 9 trisection of AnyAngle by Straightedge and Compass. Proof Of The trisection. Radius BD’ = 4. http://www.euclidchallenge.org/pg_09.htm

EUCLID CHALLENGE Successful Response by Milton Mintz May 10, 2002 Page 9: Trisection of Any Angle by Straightedge and Compass Proof Of The Trisection Radius BD’ = 4. Radius BD = 3. For any angle : The length of the arc on arc K’L’ is 4/3 of the length of the arc on arc KL. Example angle 30° Points P, P’ = 3 3/4 (1/8 of 30 Arc DT = 4/3 X 3 3/4 Arc TT’ = 2 X arc DT = 10 = 1/3 of Example Angle 30 Previous Page Top of Page Next Page

58. La Trisection De L'angle à La Règle Et Au Compas http://eleves.mines.u-nancy.fr/~taddei/Tipe/intro_tri.htm

Angles trisectables Autres trisections de l’angle p Bibliographie : , Edition Hermann, 1989; sinon, ont aussi servi comme source : Aymes,J , publication de l’APMEP Commission Inter-IREM , Ellipses, 1993 Delahaye,JP Le fascinant nombre p , Diffusion Belin, 1997 Le Lionnais Martin,GE Geometric constructions , Springer-Verlag, 1998

59. Recherche : Trisection%20d'un%20angle trisection d'un angle , Certification IDDN. Dans lesfiches. 20 fiches trouvées 2001 Bulletin de l'APMEP. http://publimath.irem.univ-mrs.fr/cgi-bin/publimath.pl?r=trisection d'un angle

60. 1995 Repères. Num. 17. P. 85-120. Le Troisième Degré En Second Cycle : Le Fil Translate this page Euler) sur la question de la résolution des équations du troisième degré parla formule de Cardan et par la voie trigonométrique (trisection de l'angle). http://publimath.irem.univ-mrs.fr/biblio/IWR97116.htm

Informations Pratiques recherche Recherche Auteur(s) : Le Goff Jean-Pierre Titre : English title : Third power in cycle two. The thread of Euler. Editeur : Notes : Bombelli Cardan Descartes Euler ... Recherche

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