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College Geometry an Introduction to the Modern Geometry of the Triangle and the Circle

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College Geometry an Introduction to the Modern Geometry of the Triangle and the Circle An Introduction to the Modern Geometry of the Triangle and the Circle Nathan AltshiLLer-Court College Geometry An Introduction to the Modern Geometry of the Triangle and the Circle Nathan Altshiller-Court Second Edition Revised and Enlarged Dover Publications, Inc. Mineola, New York Copyright Copyright © 1952 by Nathan Altshiller-Court Copyright © Renewed 1980 by Arnold Court All rights reserved. Bibliographical Note This Dover edition, first published in 2007, is an unabridged republication of the second edition of the work, originally published by Barnes & Noble, Inc., New York, in 1952. Library of Congress Cataloging-in-Publication Data Altshiller-Court, Nathan, b. 1881. College geometry : an introduction to the modern geometry of the tri- angle and the circle / Nathan Altshiller-Court. - Dover ed. p. cm. Originally published: 2nd ed., rev. and enl. New York : Barnes & Noble, 1952. Includes bibliographical references and index. ISBN 0-486-45805-9 1. Geometry, Modern-Plane. I. Title. QA474.C6 2007 5l6.22-dc22 2006102940 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 To My Wife PREFACE Before the first edition of this book appeared, a generation or more ago, modern geometry was practically nonexistent as a subject in the curriculum of American colleges and universities.Moreover, the edu- cational experts, both in the academic world and in the editorial offices of publishing houses, were almost unanimous in their opinion that the colleges felt no need for this subject and would take no notice of it if an avenue of access to it were opened to them. The academic climate confronting this second edition is radically different.College geometry has a firm footing in the vast majority of schools of collegiate level in this country, both large and small, includ- ing a considerable number of predominantly technical schools.Com- petent and often even enthusiastic personnel are available to teach the subject. These changes naturally had to be considered in preparing a new edition.The plan of the book, which gained for it so many sincere friends and consistent users, has been retained in its entirety, but it was deemed necessary to rewrite practically all of the text and to broaden its horizon by adding a large amount of new material. Construction problems continue to be stressed in the first part of the book, though some of the less important topics have been omitted in favor of other types of material.All other topics in the original edition have been amplified and new topics have been added.These changes are particularly evident in the chapter dealing with the recent geometry of the triangle.A new chapter on the quadrilateral has been included. Many proofs have been simplified.For a considerable number of others, new proofs, shorter and more appealing, have been substituted. The illustrative examples have in most cases been replaced by new ones. The harmonic ratio is now introduced much earlier in the course. This change offered an opportunity to simplify the presentation of some topics and enhance their interest. vii Viii PREFACE The book has been enriched by the addition of numerous exercises of varying degrees of difficulty. A goodly portion of them are note- worthy propositions in their own right, which could, and perhaps should, have their place in the text, space permitting.Those who use the book for reference may be able to draw upon these exercises as a convenient source of instructional material. N. A.-C. Norman, Oklahoma ACKNOWLEDGMENTS It is with distinct pleasure that I acknowledge my indebtedness to my friends Dr. J. H. Butchart, Professor of Mathematics, Arizona State College, and Dr. L. Wayne Johnson, Professor and Head of the Department of Mathematics, Oklahoma A. and M. College.They read the manuscript with great care and contributed many impor- tant suggestions and excellent additions.I am deeply grateful for their valuable help. I wish also to thank Dr. Butchart and my colleague Dr. Arthur Bernhart for their assistance in the taxing work of reading the proofs. Finally, I wish to express my appreciation to the Editorial Depart- ment of Barnes and Noble, Inc., for the manner, both painstaking and generous, in which the manuscript was treated and for the inex- haustible patience exhibited while the book was going through the press. N. A.-C. CONTENTS PREFACE . Vii ACKNOWLEDGMENTS . ix To THE INSTRUCTOR. XV To THE STUDENT. XVii 1 GEOMETRIC CONSTRUCTIONS. 1 A. Preliminaries . 1 Exercises . 2 B. General Method of Solution of Construction Problems. 3 Exercises . 10 C. Geometric Loci . 11 Exercises . 20 D. Indirect Elements . 21 Exercises . 22, 23, 24, 25, 27, 28 Supplementary Exercises. 28 Review Exercises . 29 Constructions. 29 Propositions . 30 Loci . 32 2 SIMILITUDE AND HOMOTHECY . 34 A. Similitude. 34 Exercises . 37 B. Homothecy . 38 Exercises . 43, 45, 49, 51 Supplementary Exercises . 52 3PROPERTIES OF THE TRIANGLE . 53 A. Preliminaries . 53 Exercises . 57 B. The Circumcircle . 57 Exercises . 59, 64 Xi xii CONTENTS C. Medians . 65 Exercises . .67, 69, 71 D. Tritangent Circles . .. 72 a. Bisectors . 72 Exercises . 73 b. Tritangent Centers . .. 73 Exercises . 78 c. Tritangent Radii . 78 Exercises . 87 d. Points of Contact . 87 Exercises . 93 E. Altitudes . ..... 94 a. The Orthocenter . 94 Exercises . 96 b. The Orthic Triangle . 97 Exercises . .99, 101 c. The Euler Line . 101 Exercises . 102 F. The Nine-Point Circle. 103 Exercises . 108 G. The Orthocentric Quadrilateral . 109 Exercises . 111, 115 Review Exercises . 115 The Circumcircle. 115 Medians . 116 Tritangent Circles . 116 Altitudes . 118 The Nine-Point Circle. 119 Miscellaneous Exercises . 120 4 THE QUADRILATERAL . 124 A. The General Quadrilateral . 124 Exercises . 127 B. The Cyclic Quadrilateral. 127 Exercises . 134 C. Other Quadrilaterals . 135 Exercises . 138 5 THE SIMSON LINE. 140 Exercises . 145, 149 CONTENTS Xiii 6 TRANSVERSALS . 151 A. Introductory. 151 Exercises . .. 152 B. Stewart's Theorem . .. 152 Exercises . 153 C. Menelaus' Theorem . 153 Exercises . 158 D. Ceva's Theorem .. 158 Exercises . .161,161,164 Supplementary Exercises. 165 7 H Aizmoz is DI<nsioN . 166 Exercises . .. 168, 170 Supplementary Exercises. .. 171 8CrRcLEs . .. 172 A. Inverse Points . 172 Exercises . 174 B. Orthogonal Circles. .. .. 174 Exercises . 177 C. Poles and Polars . 177 Exercises . 182 Supplementary Exercises . 183 D. Centers of Similitude. 184 Exercises . 189 Supplementary Exercises. 190 E. The Power of a Point with Respect to a Circle . 190 Exercises . 193 Supplementary Exercises. 194 F. The Radical Axis of Two Circles . 194 Exercises . 196, 200 Supplementary Exercises . 201 G. Coaxal Circles . 201 Exercises . .205, 209, 216 H. Three Circles. 217 Exercises . 221 Supplementary Exercises. 221 I. The Problem of Apoilonius . 222 Exercises . 227 Supplementary Exercises. 227 Xiv CONTENTS 9INVERSION. 230 Exercises . 242 10 RECENT GEOMETRY OF THE TRIANGLE. 244 A. Poles and Polars with Respect to a Triangle . .244 Exercises . 246 B. Lemoine Geometry. 247 a. Symmedians . 247 Exercises . 252 b. The Lemoine Point . 252 Exercises . 256 c. The Lemoine Circles . 257 Exercises . 260 C. The Apollonian Circles . 260 Exercises .. 266 D. Isogonal Lines . 267 Exercises . .269, 273 E. Brocard Geometry. 274 a. The Brocard Points . 274 Exercises . 278 b. The Brocard Circle . 279 Exercises . 284 F. Tucker Circles . 284 G. The Orthopole . 287 Exercises . 291 Supplementary Exercises. 291 HISTORICAL AND BIBLIOGRAPHICAL NOTES . 295 LIST OF NAMES . 307 INDEX . 310 TO THE INSTRUCTOR This book contains much more material than it is possible to cover conveniently with an average college class that meets, say, three times a week for one semester.Some instructors circumvent the difficulty by following the book from its beginning to whatever part can be reached during the term. A good deal can be said in favor of such a procedure. However, a judicious selection of material in different parts of the book will give the student a better idea of the scope of modern geometry and will materially contribute to the broadening of his geometrical outlook. The instructor preferring this alternative can make selections and omissions to suit his own needs and preferences.As a rough and tentative guide the following omissions are suggested.Chap. II, arts. 27, 28, 43, 44, 51-53.Chap. III, arts. 70, 71, 72, 78, 83, 84, 91-93, 102-104, 106-110, 114, 115, 130, 168, 181, 195-200, 203, 205, 211-217, 229-239.Chap. IV, Chap. V, arts. 284-287, 292, 298-305. Chap. VI, arts. 308, 323, 324, 337-344.Chap. VIII, arts. 362, 372, 398, 399, 418-420, 431, 432, 438, 439, 463-470, 479-490, 495, 499-517. Chap. IX, Chap. X, arts. 583-587, 591, 592, 595, 596, 614- 624, 639- 648, 658-665, 672-683. The text of the book does not depend upon the exercises, so that the book can be read without reference to them.The fact that it is essential for the learner to work the exercises needs no argument. The average student may be expected to solve a considerable part of the groups of problems which follow immediately the various sub- divisions of the book.The supplementary exercises are intended as a challenge to the more industrious, more ambitious student.The lists of questions given under the headings of "Review Exercises" and "Miscellaneous Exercises" may appeal primarily to those who have an enduring interest, either professional or avocational, in the subject of modern geometry. xv TO THE STUDENT The text.Novices to the art of mathematical demonstrations may, and sometimes do, form the opinion that memory plays no role in mathematics. They assume that mathematical results are obtained by reasoning, and that they always may be restored by an appropriate argument.Obviously, such' an opinion is superficial. A mathe- matical proof of a proposition is an attempt to show that this new proposition is a consequence of definitions and theorems already accepted as valid.If the reasoner does not have the appropriate propositions available in his mind, the task before him is well-nigh hopeless, if not outright impossible.
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