geometry unbound kedlaya

Download Geometry Unbound Kedlaya

Post on 23-Nov-2015

60 views

Category:

Documents

2 download

Embed Size (px)

DESCRIPTION

to the geometry lovers

TRANSCRIPT

  • Geometry Unbound

    Kiran S. Kedlayaversion of 17 Jan 2006

    c2006 Kiran S. Kedlaya. Permission is granted to copy, distribute and/or modify thisdocument under the terms of the GNU Free Documentation License, Version 1.2 or anylater version published by the Free Software Foundation; with no Invariant Sections, noFront-Cover Texts, and no Back-Cover Texts. Please consult the section of the Introductionentitled License information for further details.

    Disclaimer: it is the authors belief that all use of quoted material, such as statementsof competition problems, is in compliance with the fair use doctrine of US copyright law.However, no guarantee is made or implied that the fair use doctrine will apply to all derivativeworks, or in the copyright law of other countries.

  • ii

  • Contents

    Introduction viiOrigins, goals, and outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiMethodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiStructure of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

    I Rudiments 1

    1 Construction of the Euclidean plane 31.1 The coordinate plane, points and lines . . . . . . . . . . . . . . . . . . . . . 41.2 Distances and circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Triangles and other polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Areas of polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Areas of circles and measures of arcs . . . . . . . . . . . . . . . . . . . . . . 91.6 Angles and the danger of configuration dependence . . . . . . . . . . . . . . 111.7 Directed angle measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    2 Algebraic methods 152.1 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Barycentric coordinates and mass points . . . . . . . . . . . . . . . . . . . . 21

    3 Transformations 233.1 Congruence and rigid motions . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Similarity and homotheties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Spiral similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Complex numbers and the classification of similarities . . . . . . . . . . . . . 283.5 Affine transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    4 Tricks of the trade 334.1 Slicing and dicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    iii

  • 4.2 Angle chasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    4.3 Working backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    II Special situations 39

    5 Concurrence and collinearity 41

    5.1 Concurrent lines: Cevas theorem . . . . . . . . . . . . . . . . . . . . . . . . 41

    5.2 Collinear points: Menelauss theorem . . . . . . . . . . . . . . . . . . . . . . 43

    5.3 Concurrent perpendiculars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    6 Circular reasoning 47

    6.1 Power of a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    6.2 Radical axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    6.3 The Pascal-Brianchon theorems . . . . . . . . . . . . . . . . . . . . . . . . . 50

    6.4 Simson lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    6.5 Circle of Apollonius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    6.6 Additional problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    7 Triangle trivia 55

    7.1 Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    7.2 Incenter and excenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

    7.3 Circumcenter and orthocenter . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    7.4 Gergonne and Nagel points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    7.5 Isogonal conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    7.6 Brocard points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    7.7 Frame shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    7.8 Vectors for special points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    7.9 Additional problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

    8 Quadrilaterals 67

    8.1 General quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    8.2 Cyclic quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    8.3 Circumscribed quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    8.4 Complete quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    9 Geometric inequalities 71

    9.1 Distance inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

    9.2 Algebraic techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

    9.3 Trigonometric inequalities and convexity . . . . . . . . . . . . . . . . . . . . 75

    9.4 The Erdos-Mordell inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    9.5 Additional problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    iv

  • III Some roads to modern geometry 79

    10 Inversive and hyperbolic geometry 8110.1 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8110.2 Inversive magic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8410.3 Inversion in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8610.4 Hyperbolic geometry: an historical aside . . . . . . . . . . . . . . . . . . . . 8710.5 Poincares models of hyperbolic geometry . . . . . . . . . . . . . . . . . . . . 8810.6 Hyperbolic distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.7 Hyperbolic triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

    11 Projective geometry 9311.1 The projective plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9311.2 Projective transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9411.3 A conic section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9611.4 Conics in the projective plane . . . . . . . . . . . . . . . . . . . . . . . . . . 9811.5 The polar map and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9911.6 Cross-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.7 The complex projective plane . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    IV Odds and ends 105

    Hints 107

    Suggested further reading 111

    Bibliography 113

    About the license 115Open source for text? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Source code distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

    GNU Free Documentation License 117Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171. Applicability and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172. Verbatim Copying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193. Copying in Quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194. Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205. Combining Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226. Collections of Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227. Aggregation with Independent Works . . . . . . . . . . . . . . . . . . . . . . . 1228. Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

    v

  • 9. Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12310. Future Revisions of This License . . . . . . . . . . . . . . . . . . . . . . . . . 123Addendum: How to use this License for your documents . . . . . . . . . . . . . . 124

    Index 125

    vi

  • Introduction

    Origins, goals, and outcome

    The original text underlying this book was a set of notes1 I compiled, originally as a par-ticipant and later as an instructor, for the Math Olympiad Program (MOP),2 the annualsummer program to prepare U.S. high school students for the International MathematicalOlympiad (IMO). Given the overt mission of the MOP, the notes as originally compiled wereintended to bridge the gap between the knowledge of Euclidean geometry of American IMOprospects and that of their counterparts from other countries. To that end, they included alarge number of challenging problems culled from Olympiad-level competitions from aroundthe world.

    However, the resulting book you are now reading shares with the MOP a second mission,which is more covert and even a bit subversive. In revising it, I have attempted to usher thereader from the comfortable world of Euclidean geometry to the gates of geometry as theterm is defined (in multiple ways) by modern mathematicians, using the solving of routineand nonroutine problems as the vehicle for discovery. In particular, I have aimed