Frontmatter¶
Title, Preface, Contents, To the Instructor, To the Student
College Geometry¶
An Introduction to the Modern Geometry of the Triangle and the Circle
Copyright¶
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¶
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 educational 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, including a considerable number of predominantly technical schools. Competent 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 The illustrative examples have in most cases been replaced by new ones.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.
The book has been enriched by the addition of numerous exercises of varying degrees of difficulty. A goodly portion of them are noteworthy 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.
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 important 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 Department of Barnes and Noble, Inc., for the manner, both painstaking and generous, in which the manuscript was treated and for the inexhaustible patience exhibited while the book was going through the press.
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 |
3 PROPERTIES OF THE TRIANGLE |
53 |
A. Preliminaries |
53 |
Exercises |
57 |
B. The Circumcircle |
57 |
Exercises |
59, 64 |
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 |
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, 164 |
Supplementary Exercises |
165 |
7 HARMONIC DIVISION |
166 |
Exercises |
168, 170 |
Supplementary Exercises |
171 |
8 CIRCLES |
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 Apollonius |
222 |
Exercises |
227 |
Supplementary Exercises |
227 |
9 INVERSION |
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 subdivisions 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.
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 mathematical 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.
The student who embarks upon the study of college geometry should have accessible a book on high-school geometry, preferably his own text of those happy high-school days. Whenever a statement in College Geometry refers, explicitly or implicitly, to a proposition in the elementary text, the student will do well to locate that proposition and enter the precise reference in a notebook kept for the purpose, or in the margin of his college book. It would be of value to mark references to College Geometry on the margin of the corresponding propositions of the high-school book.
The cross references in this book are to the preceding parts of the text. Thus art. 189 harks back to art. 73. When reading art. 189, it may be worth while to make a record of this fact in connection with art. 73. Such a system of “forward” references may be a valuable help in reviewing the course and may facilitate the assimilation of the contents of the book.
Figures. The student will do well to cultivate the habit of drawing his own figures while reading the book, and to draw a separate figure for each proposition. A rough free-hand sketch is sufficient in most cases. Where a more complex figure is required, the corresponding figure in the book may be consulted as a guide to the disposition of the various parts and elements. Such practices help to fix the propositions in the reader’s mind.
The Exercises. The purpose of exercises in the study of mathematics is usually two-fold. They provide the reader with a check on his mastery of the contents of the course, and also with an opportunity to test his ability to use the material by applying the methods presented in the book. These two phases are, of course, not unrelated.
It goes without saying that the student cannot possibly solve a problem if he does not know what the problem is. To argue the contrary would be nothing short of ridiculous. In the light of experience, however, it may be useful to insist on this point. We begin our problem-solving career with such simple statements that there is no doubt as to our understanding their contents. When, in the course of time, conditions change radically, we continue, by force of habit, to assume an instantaneous knowledge of the statement of the problem.
Like the problems in most books on geometry, nearly all the problems in College Geometry are verbal problems. Nevertheless, surprising as it may sound, it is often difficult to know what a given problem is. More or less effort may be required to determine its meaning. Clearly, this effort of understanding the problem must be made first, however, before any steps toward a solution are undertaken. In fact, the mastery of the meaning of the problem may be the principal part, and often is the most difficult part, of its solution.
To make sure that he understands the statement of the question, the student should repeat its text verbally, without using the book, or, still better, write the text in full, from memory. Moreover, he must have in mind so clearly the meaning of the spoken or written sentences that he will be able to explain a problem, in his own words, to anyone, equipped with the necessary information, who has never before heard of the problem.
Finally, it is essential to draw the figure the question deals with. A simple free-hand illustration will usually suffice. In some cases a carefully executed drawing may provide valuable suggestions.
Obviously, no infallible rule can be given which will lead to the solution of all problems. When the student has made sure of the meaning of the problem, has listed accurately the given elements of the problem and the elements wanted, and has before him an adequate figure, he will be well armed for his task, and with such help even a recalcitrant question may eventually become more manageable.
The student must not expect that a solution will invariably occur to him as soon as he has finished reading the text of a question. If it does, as it often may, well and good. But, in most cases, a question requires, above all, patience. A number of unsuccessful starts is not unusual, and need not cause discouragement. The successful solver of problems is the one whose determination — whose will to overcome obstacles — grows and increases with the resistance encountered. Then, after the light breaks through, and the goal has been reached, his is the reward of a gratifying sense of triumph, of achievement.