81. Trisection Of Angles !!! - Science Forums And Debate trisection OF angleS 2ND VERSIONwhere the angle is 180 degrees or less This procedureis Similar to the first version but gives a very CLEAR PICTURE of the http://www.scienceforums.net/forums/showthread.php?goto=lastpost&t=3808

82. Xah: Special Plane Curves: Quadratrix Of Hippias It is conceived by Hippias of Ellis (ca 460 BC) to trisect the angle thussometimes called trisectrix of Hippias. Properties. Trisecting an angle. http://www.xahlee.org/SpecialPlaneCurves_dir/QuadratrixOfHippias_dir/quadratrixO

Table of Contents Quadratrix Of Hippias Mathematica Notebook for This Page History Description Formulas ... Related Web Sites History Quadratrix of Hippias is the first named curve other than circle and line. It is conceived by Hippias of Ellis (ca 460 BC) to trisect the angle thus sometimes called trisectrix of Hippias. The curve is better known as quadratrix because it is later used to square the circle. Description Step by step description: Let there be a square in the first quardrant with lower-left coner at the origin. Let there be a line parallel to the x-axis and gradually moving up at a constant speed from the bottom side of the square until it reaches the upper side of the square. Let there be an angle in standard position. The angle increase at a constant speed from to Pi/2. Both the angle and line movement start and finish simultaneously. The intersections of the line and the angle is quadratrix of Hippias. Tracing the Curve. (18 k) Tracing the Curve Formulas Polar: r == theta/Sin[theta] Cartesian: x == y Cot[y] Properties Trisecting an Angle The curve can be used to trisect any acute angle. To trisect angel AOB, one first find distance OE, take one third of distance OE to get OD (trisection of a segment can be done with ruler and compass). Let the intersection of the curve and the horizontal line at D be P, then the angle AOP is one third of angle AOB. This follows because by the definition of the curve, distance[O,E]/distance[O,C] == angle[A,O,B]/(Pi/2). If distance[O,C] == Pi/2, then distance[O,E] == angle[A,O,B].

83. Forum Trisection Angulaire angle (EGF) c est -à- dire l angle (BGF) = l angle EGF/3 3 http://www.espacemath.com/fortris2.htm

Forum Math Guide Accueil Liber Guide ... Historique Trisection angulaire N'hésitez pas à nous envoyer vos réponses afin de les insérer dans cette section. Sommaire Accueil Collège Concours Dimaf ... Rédaction Question N°1 de Sebaa Djelloul du 10/12/01 à 22h 03 : Bonjour les amis du forum, soit l'équation du troisième degré X^3 - 3 X + 1 = 0. Peut on presenter géométriquement la valeur de X à la règle et au compas sans passer par la formule de CARDAN ou les formules trigonométriques telles que (cos ou arcos). Merci Question N°2 de Sebaa Djelloul du 10/12/01 à 22h 07 : Bonjour les amis du forum, peut-on diviser un angle de 60° en trois parties égales à l'aide de la règle et du compas ? Merci Question N°3 de Sebaa Djelloul du 10/12/01 à 22h 11 : Soient deux points fixes A et B tels que AB = b. Tracer deux cercles (C1) de centre A et de rayon AB=b et (C2) de centre B et de rayon AB = b ces deux cercles se coupent en deux points C et D la droite (CD) est la médiatrice du segment AB. Soit la droite (EF)/ / (AB) tels que les points E et F appartiennnent respectivement aux cercles (C1) et (C2) Questions 1- Démontrer quelques soient les points E et F tel que la droite EF / / AB les trois médiatrices des segments EA, AB et BF se coupent en un seul point G qui est équidistant des quatres points E, A , B et F autrement dit le point G est le centre du cercle (C3) de centre G et de rayon EG = AG = BG = FG = R.

84. Introduction: trisection de l angle, duplication du cube ,l Définitions. trisection de l angle. Quadrature du cercle. Duplication du cube. http://locmant.free.fr/web/trisection_duplication_quadrature/tris_quad_dup.htm

Introduction: "Quadrature du cercle, trisection de l'angle, duplication du cube ", l'infernale trilogie responsable de nombreux tourments pour les géomètres grecs de l'époque classique, s'avéra au fil des âges, constituer comme on le sait, un extraordinaire terrain d'investigation et de découvertes. Retour a la page d'accueil Trisection de l'angle Quadrature du cercle Duplication du cube

85. Angle Trisection By Construction Of A Well-Defined Locus And A angle trisection by Construction of a WellDefined Locus and a Simple Arc. angletrisection by Construction of a Well-Defined Locus and a Simple Arc . http://www.cs.bris.ac.uk/Publications/pub_info.jsp?id=1000132

86. Flooble :: Perplexus :: Geometry : Angle Trisection To access this page on the new site, go here perplexus.info Geometry AngleTrisection (Your old login and password will still be valid on the new site). http://www.flooble.com/perplexus/show.php?pid=117

87. ComplexCurves Curve, Mathematician, Purpose Mentioned, location. cochloid or conchoid (describedby Pappus, Collectio iv §§2629), Nicomedes (PI), angle trisection (I), http://www.calstatela.edu/faculty/hmendel/Ancient Mathematics/Philosophical Text

Four passages from Athenian Neo-Platonists on complex curves translated by Henry Mendell (Cal. State U., L.A.) Return to Vignettes of Ancient Mathematics Passages: Proclus, Commentary on Euclid's Elements I Proclus, Commentary on Euclid's Elements I (p. 356.6-16) These are the same quotation from the lost work: Iamblichus, Commentary on Aristotle's Categories Simplicius, Commentary on Aristotle's Categories Simplicius, Commentary on Aristotle's Physics Curves Mentioned (P = Proclus, I = Iamblichus as quoted by Simplicius) Curve Mathematician Purpose Mentioned location cochloid or conchoid (described by Nicomedes (PI) angle trisection (I) sibling of the cochlioid Apollonius, Nicomedes (I) circle squaring (I) conics Apollonius (P) line from double motion Carpus (I) circle squaring (I) quadratrices (all sources are available) Hippias (P), Nicomedes (PI) cutting an angle in a given ratio (P), circle squaring (I) speirics (discussed by Hero, Metrica ii 13, ps.-Hero, Definitions 74-75, and Proclus) Perseus (P) none Archimedean spiral (besides Archimedes

88. Geometry: Angle Trisection This page is maintained by MathPro Press, classifier@MathProPress.com.copyright notice http://www.mathpropress.com/cmj/pages/page87.html

This page is maintained by MathPro Press classifier@MathProPress.com

89. Geometric Cryptography Identification By Angle Trisection Geometric Cryptography Identification by angle trisection (1997)(Make Corrections) Mike Burmester, Ronald L. Rivest, Adi Shamir. http://citeseer.ist.psu.edu/burmester97geometric.html

90. Constructions à La Règle Et Au Compas – Nombres Constructibles Translate this page La duplication du cube, la trisection de l’angle et la quadrature du cerclesont traitées sous un angle historique. Brochures et articles. http://mathematiques.ac-bordeaux.fr/peda/lyc/dosped/option_l/biblio_option_l.htm

Constructions à la règle et au compas Nombres constructibles Bibliographie Document d’accompagnement Théorie des corps ; La règle et le compas »  par Jean-Claude  Carréga  (Hermann 1981). Une histoire des Mathématiques routes et dédales » par Amy Dahan-Dalmedico et Jeanne Peiffer  (collection: point sciences, édition du Seuil, 1986). The mathematics of Plato’s Academy, a new reconstruction » par Fowler  (Oxford Science Publications Claredon Press 1987). L’aventure des nombres » par Gilles Godefroy  (Éditions Odile Jacob 1997). Curiosités géométriques » par Émile Fourrey  (Vuibert 2001). Un ouvrage très riche au sein duquel un chapitre est consacré à la « division des figures planes », un autre aux « applications de la géométrie au calcul ». Autres sources Les nombres et leurs mystères » par André Warusfel (collection: point sciences, édition du Seuil, 1961). Voir le chapitre : les figures régulières, sur la construction des polygones réguliers. Exercices de géométrie (comprenant l’exposé des méthodes géométriques et 2000 questions résolues) » par Frère Gabriel-Marie (Éditions Jacques Gabay).

91. Moule Des Pages Translate this page xercic e 10. 15 points. trisection d’un angle. Voici son procédé on veut faire la trisection de l’angle de la figure ci-dessous. http://www.ac-strasbourg.fr/microsites/maths_msf/1998/Exercice10.htm

x e r c i c e 10 15 points Nicomède découvrit une construction permettant de partager un angle en trois angles égaux. de la figure ci-dessous. Pour cela on a placé un point C sur le côté [Ay), on a construit la droite (d) passant par C et perpendiculaire au côté [Ax) puis la courbe conchoïde ainsi définie: pour tout point P de (d), la demi-droite [AP) coupe la courbe en M tel que PM = 2AC. La droite passant par C et perpendiculaire à d coupe la courbe en E. . Il est inutile de construire la conchoïde.

92. Hippias D'Elis 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'Élis, 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 : Trisection et Trisectrice dans ChronoMath : Hippocrate de Chio

93. Trisecting A Triangle This application enables the intrigued student to trisect angles with littledifficulty. Construction You have now trisected one angle of a triangle. http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Edenfield/Trisect/Trisectin

Trisecting a Triangle (or Morley's Theorem) By Kelly Edenfield Morley proved that given any triangle, the trisection of that triangle is always an equileral triangle. The first obstacle in proving this is the question of how to trisect the angles of a triangle. However, modern technology and Key Curriculum Press have armed us with Geometer's Sketchpad. This application enables the intrigued student to trisect angles with little difficulty. Construction: Construct an arbitrary triangle. Then measure each angle. In the calculator, divide each measure by 3. This gives the measure of one-third of the angle. Mark one of the points of the triangle as a center of rotation. Mark an adjacent edge and point at the other end of the side and highlight the one-third measure of that particular angle. Rotate by the highlighted angle. Repeat, rotating the rotation. You have now trisected one angle of a triangle. Repeat for the other two angles. Connect the trisectors. That is, join the trisectors of angle A, the trisectors of angle B, and the trisectors of angle C at their points of intersection with trisectors of other angles. This will yield a triangle. Two triangles are possible. The desired triangle is composed of parallels to the sides of the original triangle. (In the following sketch, the blue triangle.) Measure the sides of this new triangle using the measure length function. All sides are equal; therefore, this new triangle is an equilateral triangle. Move the points of the arbitrary triangle around. The new triangle will always be an equilateral.

94. Trisecting The Angle Trisecting the angle. The ancient Greeks conjectured about it. Modern mathematiciansproved that it can t be done. Use the line segments to trisect the angle. http://www.geocities.com/CapeCanaveral/Launchpad/5577/musings/trisect.html

Trisecting the Angle The ancient Greeks conjectured about it. Modern mathematicians proved that it can't be done. Here's an interesting approximation. Background : In high-school, I conjectured that if you could trisect a line, you could use that trisection to trisect an angle. Where did I make my error? The Method: Step 1: Start with an angle and draw a line across that angle. Step 2: Triple that line by replicating it twice. (Replication of a line is a well known construction for which all the steps will not be shown.) Step 3: Translate this new line, using perpendiculars, to where its ends meet the angle. Step 4: You now have a trisected line crossing an angle. Use the line segments to trisect the angle. Since we know it can't be done, what is the error in our method? It is our original assumption that by trisecting a line we can trisect an angle. If we show the lines that really must be equal to trisect an angle we can show that our method is only an approximation. Challenge for students: How good of an approximation is this? Method 2: Why can't we use the tripling idea with angles instead of lines?

95. MathsNet: Geometric Construction Course - Classic Problems Of Geometry lassic problems classicTo trisect the angle CAB Archimedes method This method isusually attributed to Archimedes (born 287 BC in Syracuse, Sicily, died 212 http://www.mathsnet.net/campus/construction/classic3.html

Thinking Classic Tech lassic problems To trisect the angle CAB Archimedes method This method is usually attributed to Archimedes (born: 287 BC in Syracuse, Sicily, died: 212 BC in Syracuse, Sicily) and uses the principal of Archimedes' spiral. Please enable Java for an interactive construction (with Cinderella). Please enable Java for an interactive construction (with Cinderella). Please enable Java for an interactive construction (with Cinderella). The conchoid of Nicomedes This second method is usually attributed to Pappus (born: 8 Feb 411 in Constantinople (now Istanbul), Byzantium (now Turkey), died: 17 April 485 in Athens, Greece) and is based on a curve known as the conchoid of Nicomedes (born: about 280 BC in Greece, died: about 210 BC). Pappus wrote: Nicomedes trisected any rectilinear angle by means of the conchoidal curves, the construction, order and properties of which he handed down, being himself the discoverer of their peculiar character Please enable Java for an interactive construction (with Cinderella).

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