61. Angles Orientés De Vecteurs Dans Le Plan Translate this page that the reflected points with respect to the sides of a triangle orthocenter areon its circumscribed circle, - the Simson's theorem - the napoleon's theorem. http://coq.inria.fr/contribs/Angles.html

Angles orientés de vecteurs dans le plan A partir d'une axiomatisation des angles orientés de vecteurs du plan euclidien,on donne des preuves classiques des théorèmes de cocyclicité, de Simson, de Napoléon et de l'orthocentre. Voir le rapport de recherche associé (http://www-sop.inria.fr/lemme/FGRR.ps) et le fichier README. Download (archive compatible with Coq V7.4) Author: Frédérique Guilhot (Frederique.Guilhot@sophia.inria.fr) Institution: INRIA Sophia Antipolis, projet Lemme Date: 15 janvier 2002 Keywords: Pcoq géométrie théorème démonstration angle cercle geometry theorem proof angle circle The README file of the contribution: This page was automatically generated from this description file

62. NRICH | Interactivities Archive Explorer only); napoleon's theorem December 1998; Thébault's Theorem- November 1998; The Eyeball Theorem - October 1998; Chords http://nrich.maths.org/mathsf/journalf/rb_interact_geom.html

Interactivities from the archive NRICH Prime NRICH Club ... Get Printable Page March 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News You will be able to access the problems in the archive using the links below: Java enabled to get the interactive diagram. Mathematical Challenges Proof Sorters - May, June, Sept 2001 Muggles Magic -June 2001 Gnomon I -May 2001 Nine Colours -April 2001 ... The Mathemagical Mystery Tour - March 1999 Frieze Factory Escher's Little Tricks Grow a Tree Two Stones - February 1999 Daisy - November 1998 Dynamic Geometry Problems Rhombus in Rectangle - March 2001 Triangle Incircle Iteration - December 2000 Circle Angles - October 2000 Angle Trisection - September 2000 ... Quads - January 1998

63. NAPOLEON BONAPARTE Coxeter and Greitzer then remark that Napoleon probably did not know enough geometryto discover napoleon's theorem, just as he probably did not know enough http://faculty.evansville.edu/ck6/bstud/napoleon.html

Napoleon Bonaparte (1769-1821) Emperor of the French The famous Napoleon Theorem is stated by Coxeter and Greitzer as follows: If equilateral triangles are erected externally on the sides of any triangle, their centers form an equilateral triangle. They continue with a historical anecdote: It is known that Napoleon Bonaparte was a bit of a mathematician with a great interest in geometry. In fact, there is a story that, before he made himself ruler of the French, he engaged in a discussion with the great mathematicians Lagrange and Laplace until the latter told him, severely, "The last thing we want from you, general, is a lesson in geometry." Laplace became his chief military engineer. Coxeter and Greitzer then remark that Napoleon probably did not know enough geometry to discover Napoleon's Theorem, just as he probably did not know enough English to compose the palindrome often attributed to him: Able was I ere I saw Elba. The portrait is by Anne-Louis Girodet-Trioson (1767-1824). I thank the MAA for permission to quote from H. S. M. Coxeter and S. L. Greitzer

64. CONTRIBUTED PRESENTATIONS SCHEDULE Gary Richter, Southwestern University. SESSION II, 253 Maguire 230 245 ASquare Version of napoleon's theorem; Bo Green, Abilene Christian University; http://www.tamu-commerce.edu/AcademicOrganizations/maa/papers98.html

CONTRIBUTED PRESENTATIONS SCHEDULE Friday Afternoon, March 27, 1998 Sessions will be in the Cox School of Business SESSION I 251 Maguire A Tangent-Secant Method for Finding the Roots of Differences of Concave Functions Ronald Prather, Trinity University (I)Estimating n! Quickly (II) Generalized Secant Method On the Numerical Verification of the Asymptotic Expansion of Duffing's Equation Shishen Sam Xie, University of Houston-Downtown A Series, A Double Integral, and Symmetry Alan Wiederhold, San Antonio College When Can an Optimization Problem be Solved by Sorting? Frank Mathis, Baylor University Computer Vision and y to the x = x to the y Tim Donovan, Midwestern University Potential and Consistency for Semivalues of Cooperative Games With Transferable Utilities Irinel Dragan, University of Texas at Arlington Derivatives without Limits Gary Richter, Southwestern University SESSION II 253 Maguire A Square Version of Napoleon's Theorem Bo Green, Abilene Christian University Erdos-Mordell Inequality in Minkowski Geometry Mostafa Ghandehari, Texas Christian University

65. Math5337: Final Projects Spring 1995. napoleon's theorem. Mark Hylden Johanna Knutson Iva Nelson SethPeterson. Tessellations. Cecilia Donarski Bob Hazen Kristin Lee Aki Yoshino. http://www.geom.umn.edu/~demo5337/

Math 5337: Technology in the Geometry Classroom Homepage Student Projects from Math 5337 Computational Methods in Elementary Geometry (Technology in the Geometry Classroom) Spring 1997 Symmetry Maryann Dorsey Kerry Hanifl Gary Kannel ... Marc Schwach Summer 1996 Classification of Patterns Kari Brown-Herbst - Unalaska City Schools, Unalaska, AK Kira Donnelly - Westridge School, Pasadena, CA Richard Stratton - Mann Middle School, Colorado Springs, CO Geri Anderson-Nielsen - Georgetown Day School, District of Columbia Concurrency Points in a Triangle Melkana Brakalova - Hotchkiss Academy, Lakeville, CT Gloria Johnson - Hospital/Homebound, Orange County Schools, FL Larry Scalzitti - Adlai E. Stevenson High School, Lincolshire, IL Linda Thornberry - Newport High School, Bellevue, WA The Pythagorean Theorem Sandra Hightower - Skyline High School, Dallas, TX Troy Jones - Richfield High School, Richfield, UT Eddie Keel - TN Virginia Urban - Fashion Institute of Technology, New York, NY Geometry In Art Workshop for the Minneapolis Public Schools Richard Dehlinger - South High School, Minneapolis, MN

66. RR-4362 : Proofs With Coq Of Theorems In Plane Geometry Using Oriented Angles. P Translate this page that the reflected points with respect to the sides of a triangle orthocenter areon its circumscribed circle, the Simson's theorem and the napoleon's theorem. http://www.inria.fr/rrrt/rr-4362.html

RR-4362 - Proofs with Coq of theorems in plane geometry using oriented angles. pages. Les rapports de cet auteur Rapport de recherche de l'INRIA- Sophia Antipolis Page d'accueil de l'unité de recherche Fichier PostScript / PostScript file Fichier postscript du document : 711 Ko Fichier PDF / PDF file Fichier PDF du document : 511 Ko Projet : LEMME - 23 pages - Janvier 2002 - Document en anglais Page d'accueil du projet Abstract : Formalization of the theory of oriented angles of non zero vectors using Coq is reported. Using this theory, some classical plane geometry theorems are proved, among them : the theorem which gives a necessary and sufficient condition so that four points are cocyclic, the one which shows that the reflected points with respect to the sides of a triangle orthocenter are on its circumscribed circle, the Simson's theorem and the Napoleon's theorem. Elaboration of proofs using Coq that followed the traditional proofs in geometry, and the difficulties encountered are described. Use of the interface Pcoq allows notations close to mathematical ones. KEY-WORDS : COQ / PCOQ / GEOMETRY / THEOREM / PROOF / ANGLE / CIRCLE

67. Microteaching Presentations Box Problem; Michelle Wang Dijkstra's Algorithm. 2000 Chrissy Folsom- A Geometric Proof of napoleon's theorem; Rob Guzzo - Parametric http://www.math.ucla.edu/~tat/microteach.html

Mathematics 495B: Teaching and Technology Spring Quarter 2002 Microteaching Presentations You can view presentations from previous years here: Roya Furmuly - Cryptography Julia Lucas - Average Value of a Function Narineh Movsessian - The Cereal Box Problem Michelle Wang - Dijkstra's Algorithm ... Richard Tsai - Radix Sort using Queue

68. Mathematical Resources: GEOMETRY (Math Links By Bruno Kevius) Michael Thwaites' Ellipses; Monge's Theorm and Desargues' Theorem,Identified; napoleon's theorem Geometry Forum, Swarthmore College; http://www.abc.se/~m9847/matre/geometr.html

Mathematical Resources Mathematics Links by Bruno Kevius This list is continually under development Geometry Adventitious Angles, Triangles and Quadrangles Angle Trisection by Bob Hesse Geometry Forum Articles Articles in "geometry.institutes" Thomas B anchoff's Project List ... Basic Terms Math League The Book "A=B" Build a Rainbow (Gallery of Interactive geometry) Cabri II Conic Macros By Jim King, U. of Washington. Cartesio 3.03e Geometry - Projections - CAD - Education - Freeware Circles of Light: The Mathematics of Rainbows (Gallery of Interactive Geometry) Center for Geometry Analysis Numerics and Graphics (GANG) Center for Hypergeometric Systems/WWW Department of Mathematics, Kobe , University Rokko College Algebra and Geometry Notes - UWMC Prepared by M. Maheswaran of the University of Wisconson - Marathon Center (UWMC) Computational Geometry Resources Carleton Computer Science Graduate Society Conic Sections (Visual Dictionary of Special Plane Curves) - Xah Lee Connected Geometry Education Development Center, Inc. Constant Perimeter and Area Rectangles Course Materials from the Geometry Center Educational Materials: The University of Minnesota CSC Mathematical Topics: Visualizations Dave's Math Tables Deep Secrets - The Great Pyramid, The Golden Ratio and The Royal Cubit

69. Gleichseitiges Dreieck Translate this page Alexander Bogomolny (Cut The Knot!) napoleon's theorem (A proof by tesselation,A proof with complex numbers, A second proof with complex numbers, Two proofs http://www.mathematische-basteleien.de/dreieck.htm

Gleichseitiges Dreieck Inhalt dieser Webseite Was ist ein gleichseitiges Dreieck? Formeln zum Dreieck Ein Punkt im Dreieck Quadrat und Dreieck Vierecke im Dreieck ... Beweise "Mathematische Basteleien" Was ist ein gleichseitiges Dreieck? Wie der Name sagt, ist das gleichseitige Dreieck ein Dreieck mit gleich langen Seiten. Wenn auf dieser Seite vom Dreieck die Rede ist, so ist das gleichseitige Dreieck gemeint. Formeln zum Dreieck top Es gilt damit R = (2/3)*h = (1/3)*sqr(3)*a und r = (1/3)*h=(1/6)*sqr(3)*a. Beweis: ED ist Mittellinie und parallel zu AB. Es gilt nach dem zweiten Strahlensatz: MA:MD=AB:ED. Daraus folgt mit AB=a und ED=a/2 die Beziehung MA MD=2 1, qed. Ein Punkt im Dreieck top Beweis am Ende! Zeichnet man von einem Punkt im Dreieck aus die Lote auf die Seiten und die Verbindungslinien zu den Eckpunkten, so entstehen sechs Dreiecke. Beweis am Ende! 3-4-5-Punkt Quadrat und Dreieck top Es sei a die Seite des Dreiecks und b die Seite des Quadrates. Diese Formel leitet man mit Hilfe des Strahlensatzes (blau) und den Beziehungen h = (1/2)*sqr(3)*a und b = sqr(2)*x her. Vierecke im Dreieck top Dreieck(e) im Dreieck top Gedrehtes und gestauchtes Dreieck Beweisgang am Ende!

70. INVESTIGATING HISTORICAL PROBLEMS centers of the circles. Figure 5 napoleon's theorem. Other questionsor extensions of this construction could be · What happens if http://www.ma.iup.edu/MAA/proceedings/vol1/enderson/enderson.htm

INVESTIGATING HISTORICAL PROBLEMS USING GEOMETER'S SKETCHPAD Mary C. Enderson Indiana University of Pennsylvania, Mathematics Department 233 Stright Hall, Indiana, PA 15705 Investigating Historical Problems Using Geometer's Sketchpad Mary C. Enderson Naturally, history has a place in the mathematics classroom that should not be overlooked. What many mathematicians fail to recognize is the enhancement of historical investigations by use of technology. Geometer's Sketchpad , a dynamic and interactive piece of software, provides a work environment that allows one to create, test, validate, and manipulate objects. It has the power and flexibility to allow students to examine an infinite number of situations, instead of one singular static case, which is invaluable in attempts to make mathematical conjectures and generalizations. The purpose of this paper is not to shed new light on tasks or problems related to history of math, but to share "golden" opportunities where use of Geometer's Sketchpad (GSP) enhances the investigation of many famous geometric problems. The scope of situations to investigate with this software are unlimited. Users quickly see how technology often generates many additional questions or tasks for students to explore, as well as enabling them to visualize the connections among various mathematics topics.

71. InterMath | Investigations | Algebra | Graphing Related External Resources. napoleon's theorem This exploration examinesthe areas of triangles formed by centroids of different triangles. http://www.intermath-uga.gatech.edu/topics/algebra/graphing/r10.htm

Search the Site Investigations Algebra Graphing ... Recommended Investigations Create a triangle and the medians of the triangle. How do the coordinates of the centroid (the intersection of the triangle's medians) relate to the coordinates of the vertices? Explain how you derived your answer. Related External Resources Napoleon's Theorem This exploration examines the areas of triangles formed by centroids of different triangles. http://www-math.uncc.edu/~droyster/courses/fall96/maed3105/gsproject 3.html Centroid This interactive investigation shows the ratio of the divided medians at the centroid. [ java ] http://www.keypress.com/sketchpad/java_gsp/centroid.html Geometer's Sketchpad Triangle Medians This worksheet can be used as a set of instructions to construct, measure, and record data related to the divided medians of a triangle. http://www.geocities.com/SoHo/Lofts/3232/trimedia.html Submit your idea for an investigation to InterMath

72. Professur Martini - Publikationen 180. MR 99 j 52016 H. Martini, B. Weissbach napoleon's theorem withweights in nspace. Geometriae Dedicata 74 (1999), 213-223. http://www.mathematik.tu-chemnitz.de/prof/mart_pap.html

Professur Geometrie Prof. Dr. Horst Martini Publikationen Martini,H. / Soltan, V: Primitive inner illuminating systems of convex bodies. Geom. Dedicata 80 (2000), 81-97. Kupitz, Y./ Martini, H: On the isoperimetric problem for Reuleaux polygons. Journal of Geometry 68 (2000), 171-191. Kupitz, Y./ Martini, H: On the equation x y = y x . Elemente der Mathematik 55 (2000), 95-101. Kupitz, Y./ Martini, H: On the weak circular intersection property. Studia Sci. Math. Hungar. 36 (2000), 371-385. Weiss,G. / Martini, H.: Heinz, G.: Lower and upper bounds on the total weight of semi-rich acyclic arrangements of oriented lines in the plane. Discrete Math. 219 (2000), 107-122. Heinz, G.: Wenzel, W. / Ay, N. / Pasemann, F: Hyperplane arrangements separating arbitrary vertex classes in n -cubes. Advances Appl. Math. 25 (2000), 284-306. V. Boltyanski, H. Martini, V. Soltan: Geometric Methods and Optimization Problems. Monograph, 429+vi pp., Kluwer Academic Publishers, Dordrecht-Boston-London, 1999. H. Martini, P. S. Soltan: On convex partitions of polygonal regions. Discrete Mathematics 195 (1999), 167-180. MR 99 j: 52016

73. Writing Assignment #4: Technology Applications that are appropriate for an interactive approach, including applications of Menelausand Ceva's theorem, Steiner's theorem, napoleon's theorem, problems with http://www.math.ilstu.edu/~day/courses/old/326/wa04sample.html

Illinois State University Mathematics Department MAT 326: Technology Tools for Secondary School Mathematics Dr. Roger Day ( day@math.ilstu.edu Technology Applications for the Classroom: A Sample Report Roger Day return to Writing Assignment #4 a) McGehee, Jean J. "Interactive Technology and Classic Geometry Problems." Mathematics Teacher 91 (March 1998): 204-208. b.i) dynamic geometry software Geometer's Sketchpad b.iii) The author compares two approaches, a traditional approach and an interactive approach, for using dynamic geometry software to explore the circle of Appolonius. She provides step-by-step instructions on both approaches that a Sketchpad user can follow. She claims that the differences in approaches focus on whether students are provided any opportunity to investigate, conjecture, and otherwise carry out some of the steps that a mathematician may actually undergo in attempting to solve a problem. The traditional approach results in a successful verification of the constant ratio in the circle of Appolonius, but allows little if any investigation by users as well as fostering little connection between the concepts involved and the construction carried out. The interactive approach allows users to first experiment and carry out many examples of the situation in order to discover the resultthe constant ratioas a result of the construction. This seemingly subtle difference, the author contends, spells the difference between students simply following and completing a procedure to focusing on the concept of the locus and using technology for exploration and discovery. The author provides suggestions of other classical geometry constructions that teachers might consider for similar interactive approaches. In so doing, students and teachers will experience more completely the kind of activities engaged in by mathematicians.

74. Projects Pick's theorem; Fibonnaci numbers; hyperbolic geometry; Ptolemy's theorem;Heron's formula; napoleon's theorem; Archimedes; Descartes; taxicab http://www.math.uga.edu/~clint/2002/5200/projects.htm

Geometry term projects Projects are due Tuesday, December 3. Project proposals are due Tuesday, October 8. Projects may be done by a group (of no more than 4 people) or individually. The format of the project may be a paper, a class presentation, a web site, GSP files, or any combination of these things. Your class presentation may involve computer, video, or a class activity. I am very open to different methods of presentation. Part of your proposal is to define the group and the format. You may not do geometry lesson plans as your project unless you have teaching experience. Good starting points for projects are the references listed on the course home page: Useful texts Yahoo! Geometry Math Forum Geometry bibliography . There is a wealth of geometry on the internet! I own many geometry books, and I will lend them to you. Here are some suggestions for topics, in no particular order. You may go into the history, focus on specific theorems, or combine both viewpoints. You may talk about applications or relations to other parts of mathematics such as algebra or calculus. Pythagorean theorem squaring the circle, trisecting an angle

75. Journal For Geometry And Graphics 5, No. 1 (2001) 1, 2001 · Contents. Akos G. Horvath, Istvan Prok Packing Congruent Bricks intoa Cube. Pavel Pech The Harmonic Analysis of Polygons and napoleon's theorem. http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JGG/5.1/

Journal for Geometry and Graphics Akos G. Horvath, Istvan Prok: Packing Congruent Bricks into a Cube Pavel Pech: The Harmonic Analysis of Polygons and Napoleon's Theorem Naomi Ando, Nobuhiro Yamahata, Syuta Masumi, Masahiro Chatani: Shape Grammar and Form Properties of Architectural Figures Kiyoe Fuchigami: Analysis of the Spiral Pattern Karakusa Kazuya Kojima, Mikiya Hironaga, Sadahiko Nagae, Yasuhiro Kawamoto: A Human Motion Analysis Using the Rhythm - A Reproducing Method of Human Motion Kazuko Mende: Light and Shadow in Painting - Concerning the Expression of Shadows in Western Painting - Takafumi Noguchi, Yoshio Ohno: A Deformation Algorithm of Railway Maps Michikazu Ohnishi: A Photographic Method for Panoramic Sequence with a Regular Camera Part 3: Application to Sky Photographs International Experiences in Developing the Spatial Visualization Abilities of Engineering Students A Proposal for an On-Line Library of Descriptive Geometry Problems Emiko Tsutsumi, Ayane Ichikawa, Nana Kadowaki: Evaluation of Mentally Perceived Differences Between the 3D Objects Used in Mental Cutting Tests Journal Home All Journals ELibM Home ... EMIS Home Publication date for the electronic files: 29 Aug 2001. Last modified: 29 August 2001. ELibM for the EMIS Electronic Edition

76. Andrew Glassner's Notebook Mirror reflections and billiard balls give way to mathematical constructs suchas Ptolemy's Theorem, napoleon's theorem and Fourier transformations. http://www.glassner.com/andrew/writing/books/notebook.htm

Andrew Glassner's Notebook In 1996 I started writing a regular column for the magazine IEEE Computer Graphics and Applications . I'm happy with my columns, but there are often things that need to get cut out for space reasons. Sometimes I realize some things could have been done better. And errors do make it into print. I've collected the first three years of columns, restored them to their original full-length form, expanded and revised each one, and fixed the errors, resulting in this book. The book now has a sequel, Andrew Glassner's Other Notebook The idea that graphics is fun is reflected in the book's subtitle, Recreational Computer Graphics . The cover is a notebook-style collage of some illustrations from different chapters, evoking the idea of a notebook. You can read notes on the original columns, plus the ones that haven't yet been collected

77. Standard 5 Students Use A Variety Of Techniques And Tools To triangle theorem, a theorem using the Pythagoras diagram but yielding a surprisingtriangle, a derived quadrilateral, and napoleon's theorem); and mechanical http://arachne.rfsd.k12.co.us/Standards/RoaringForkWork/7thmath-standard5.htm

th Grade Web Links Organized by Standards Standard V Standard 5: Students use a variety of techniques and tools to measure, apply the results in problem solving situations, and communicate the reasoning used in solving these problems. Click Here (5-7) Geometry; Discovery of Pi (6-8) Metric Conversion Method (4-8) Measurement 7-8 Areas and Volumes Grade : 7 estimation, feet, table, rate, distance, formula, yards, points, addition, subtraction Measuring Distance in Cocowalk 7th Grade length, width, area THE MORIKAMI AREA LESSON grade 4-8 SMILE METRIC STYLE use a metric ruler 4-7 MEASURING MEASUREMENT 7-11 "ANYWHERE IN THE USA"distance formula and adding, and subtracting decimals. MS It's a matter of density (math-science linked) 6-12 Polyominoes - Students use two famous games, dominoes and tetris to explore a whole class of geometric figures. Students will explore perimeter, and use patterns to graph the perimeter function. Middle School Rectangle Pattern Challenges Students look for patterns and determine formulas from a geometric design. Middle School General math, number notation and other math categories are taught for middle school students.

78. Credits Credits. Special thanks to graphicindex. Math Forum What Is a Tessellation.napoleon's theorem. Google Search tesselation. Math Cats make tessellations. http://www.cvcaroyals.org/runyon/credits.htm

Credits Special thanks to: graphicindex Math Forum What Is a Tessellation Napoleon's Theorem Google Search tesselation ... Teacher Page

79. Brianchon before someone realised that its dual, which is Brianchon's theorem, would also theEcole Polytechnique and became a lieutenant in artillery in napoleon's army http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Brianchon.html

80. Napoleon Theorem The summary for this English page contains characters that cannot be correctly displayed in this language/character set. http://140.114.32.3/summer99/18/work14.html

Napoleon's Theorem Illustrate Napoleon's Theorem : the centers L, M, N of the three equilateral triangles DBXC, DCYA, DAZB built outwards on the sides BC, CA, AB of an arbitrary triangle DABC are the vertices of an equilateral triangle. The same is true of the centers of the three inward equilateral triangles. Generalize Napoleon's Theorem. (a) Explore interesting porperties associated with the Torricelli's configuration. (¤T¹ïÀ³äªº¤¤ÂI³s¦¨¥¿¤T¨¤§Î) (b) Explore interesting porperties associated with the Torricelli's configuration. ( ¤T­Ó¤p¤T¨¤§Îªº­«¤ß³s¦¨¥¿¤T¨¤§Î) (c) Explore interesting porperties associated with the Torricelli's configuration. (d) Explore interesting porperties associated with the Torricelli's configuration. (¤T±øª½½u¦@ÂI) Explore interesting porperties associated with the Torricelli's configuration. (¥H¤T¨¤§Îªº¤Tä§@¤T­Ó¬Û¦ü¤T¨¤§Î) (e) Explore interesting porperties associated with the Torricelli's configuration. (¤T­Óºñ¦â¤T¨¤§Îªº¥~¤ß³s¦¨¬Û¦ü¤T¨¤§Î) (f) Explore interesting porperties associated with the Torricelli's configuration.

Page 4     61-80 of 90    Back | 1  | 2  | 3  | 4  | 5  | Next 20