Euclidean Geometry in Mathematical Olympiads

Description

This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral. The exposition is friendly and relaxed, and accompanied by over 300 beautifully drawn figures. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains as selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions. This book is especially suitable for students preparing for national or international mathematical olympiads, or for teachers looking for a text for an honor class. Content: Preface Preliminaries Part I. Fundamentals: 1. Angle chasing 2. Circles 3. Lengths and rules 4. Assorted configurations Part II. Analytic Techniques: 5. Computational geometry 6. Complex numbers 7. Barycentric coordinates Part III. Farther from Kansas: 8. Inversion 9. Projective geometry 10. Complete quadrilaterals 11. Personal favorites Part IV. Appendices: Appendix A. An ounce of linear algebra Appendix B. Hints Appendix C. Selected solutions Appendix D. List of contests and abbreviations Bibliography Index About the author.