Euclidean Geometry in Mathematical Olympiads

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Euclidean Geometry in Mathematical Olympiads Evan Chen, 2016 ==================== 奥利匹克竞赛中的几何学 英文原版,书签带目录。
⊙2016by The Mathematical association of america(Incorporated) Library of Congress Control Number: 2016933605 Print isbn:978-0-88385-8394 Electronic ISBN: 978-1-61444-411-4 Printed in the united states of america Current Printing (last digit) 10987654321 Euclidean Geometry in Mathematical olympiads With 248 ustrations Evan chen E MAA Published and Distributed by The mathematical association of america Council on publications and Communications Jennifer J. Quinn, Chair Committee on books Jennifer J. Quinn, Cha MAA Problem books editorial board Gail s Nelson edito Claudi alsina Scott annin Adam. berliner Jennifer roche bowel Douglas b. meade John h. rick Zsuzsanna szaniszlo Eric r. Westlund MAA PROBLEM BOOKS SERIES Problem books is a series of the Mathematical Association of America consisting of collections of problems and solutions from annual mathematical competitions; compilations of problems (including unsolved problems) specific to particular branches of mathematicS; books on the art and practice of problem solving, etc Ahal Solutions. Martin Erickson The Alberta High school Math Competitions 1957-2006: A Canadian Problem book, compiled and edited by andy liu The Contest Problem Book vll: American Mathematics CompetitionS, 1995-2000 Contests compiled and augmented by harold B. reiter The Contest Problem book vlll: American Mathematics Competitions(AMC 10), 2000- 2007, compiled and edited by j. Douglas Faires david Wells The Contest Problem Book IX: American Mathematics Competitions(AMC 12), 2000-2007, compiled and edited by david wells &j. douglas faires Euclidean geometry in Mathematical Olympiads, by Evan Chen First Steps for Math Olympians: Using the American Mathematics Competitions, by J Douglas faires A Friendly Mathematics Competition: 35 Years of Teamwork in Indiana, edited by rick Gillman A Gentle Introduction to the American Invitational Mathematics Exam, by Scott A. Annin Hungarian Problem Book /v, translated and edited by robert Barrington Leigh and Andy Liu The Inquisitive Problem solver, Paul Vaderlind, Richard K Guy, and Loren C. Larson International Mathematical Olympiads 1986-1999, Marcin E. Kuczma Mathematical Olympiads 1998-1999: Problems and Solutions From Around the World, edited by Titu Andreescu and Zuming Feng Mathematical Olympiads 1999-2000: Problems and Solutions From Around the World, dited by Titu Andreescu and Zuming Feng Mathematical Olympiads 2000-2001 Problems and Solutions From Around the World, edited by Titu Andreescu, Zuming Feng, and George Lee, Jr A Mathematical Orchard: Problems and Solutions, by Mark I. Krusemeyer, George T Gilbert. and Loren c. larson Problems from Murray klamkin: The Canadian Collection, edited by andy liu and bruce Shawyer Trigonometry: A Clever Study Guide, by james Tanton The william Lowell Putnam Mathematical Competition Problems and Solutions:1938- 1964.A M. Gleason. R. E. Greenwood. L M. Kell The william Lowell Putnam Mathematical Competition Problems and Solutions:1965 1984. Gerald L. Alexanderson Leonard F Klosinski and loren C. larson The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, Kiran S. Kedlaya, Bjorn Poonen, Ravi vakil USA and International Mathematical Olympiads 2000, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2001, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2002, edited by Titu Andreescu and Zuming fen g USA and International Mathematical Olympiads 2003, edited by Titu Andreescu and Zuming Feng USA and International Mathematical olympiads 2004, edited by Titu Andreescu, Zuming Feng, and po-shen loh MAA Service center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-lMAA FAX:1-240-396-5647 Dedicated to the Mathematical Olympiad Summer Program Contents Preface Preliminaries 0.1 The Structure of This book 0.2 Centers of a Triangle XIV 0. 3 Other notations and conventions XV Fundamentals 1 Angle Chasing 3 1. 1 Triangles and Circles........... 1.2 Cyclic Quadrilaterals 6 1. 3 The Orthic Triangle 1. 4 The Incenter/Excenter lemma 1.5 Directed Angles 1.6 Tangents to Circles and Phantom Points 15 1. 7 Solving a problem from the imo shortlist ..16 1. 8 Problems 18 Circles 23 2. 1 Orientations of Similar Triangles 23 2.2 Power of a point 24 2. 3 The Radical axis and radical center 26 axia 30 2.5 Revisiting Tangents: The Incenter 31 2. 6 The excircles 32 2.7 Example Problems 34 2.8 Problems 39 3 Lengths and ratios 43 3.1 The Extended law of sines 43 3.2Ceva’ s Theorem 44 3.3 Directed lengths and menelaus Theorem 46 3.4 The Centroid and the medial Triangle 48 Contents 3.5 Homothety and the Nine-Point Circle 49 3.6 Example Problems 51 3.7 Problems 56 Assorted Configurations 59 4.1 Simson Lines revisited 4.2 Incircles and excircles 4.3 Midpoints of altitudes 62 4.4 Even More Incircle and Incenter Configurations 4.5 Isogonal and Isotomic Conjugates 63 4.6S 4.7 Circles inscribed in segments 66 4. 8 Mixtilinear incircles 68 4.9 Problems Ii Analytic Techniques.....................73 5 Computational Geometry 75 5.1 Cartesian Coordi 75 5.2A 77 5.3 TrigonometI 79 5.4 Ptolemys Theorem 81 5.5 Example problems 84 5.6 Problems omplex Numbers 与 6. 1 What is a Complex Number? 95 6.2 Adding and Multiplying Complex Numbers 96 6.3 Collinearity and Perpendicularity 6.4 The Unit circle 100 6.5 Useful Formulas 6.6 Complex Incenter and Circumcenter ..106 6.7 Example Problems ...108 6.8 When(Not)to use Complex Numbers 6.9 Problems 7 Barycentric Coordinates 119 7.1 Definitions and first Theorems 119 7.2 Centers of the Triangl 122 7.3 Collinearity, Concurrence, and points at Infinity 123 7.4 Displacement Vectors ..126 7.5 A Demonstration from the imo shortlist 129 7.6 Conway's notations ....132 7.7 Displacement Vectors, Continued 7.8 More Examples 135 7.9 When(Not) to Use Barycentric Coordinates 142 7.10 Problems ....143 Contents X I Farther from Kansas 。147 8 Inversion 149 8.1 Circles are lines 149 8.2 Where do clines go? 15 8.3 An Example from the USaMO 154 8.4 Overlays and orthogonal circles 156 8.5 More overlay 159 8. 6 The Inversion distance formula ....160 8.7 More Example Problems ..160 8. 8 When to Invert 165 8. 9 Problems 165 9 Projective Geometry 169 9. 1 Completing the Plane 169 9.2 Cross ratios ∴....170 9.3 Harmonic Bundles 173 9. 4 Apollonian Circles 176 9.5 Poles/Polars and Brocard's Theorem ...178 9.6 Pascals Theorem 181 9.7 Projective Transformations 183 9.8 Example 185 9.9 Problems 10 Complete Quadrilaterals 195 10.1 Spiral similarity 196 10.2 Miquel's Theorem 198 10.3 The Gauss-Bodenmiller Theorem 198 10.4 More Properties of general miquel points 200 10.5 Miquel Points of Cyclic Quadrilaterals 201 10.6 Example problems 202 10.7 Problems 205 11 Personal Favorites 209 IV Appendices 213 Appendix A: An Ounce of Linear Algebra 215 A1 Matrices and determinants ..215 A 2 Cramer's rule 217 A 3 Vectors and the dot product .........217 Appendix B: Hints 221 Appendix C: Selected Solutions 241 C1 Solutions to Chapters 1-4 ....241 C 2 Solutions to Chapters 5-7 251

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