Geometry A High School Course

A geometry course based on this book was taught success­ fully by Gene Murrow for several years. We are much indebted to Springer-Verlag for publishing Geometry, so that others can try our approach. The publishers and we thought it would be appropriate to

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Geometry A High School Course With 545 Illustrations

Springer Science+Business Media, LLC

Serge Lang

Gene Murrow

Department of Mathematics Yale University New Haven, CT 06520 USA

1 Farm Road Ardsley, NY 10502 USA

AMS Classifications: 51-01, 51-M05, 00-01

Library of Congress Catalog ing in Publication Data Lang, Serge, 1927Geometry: a high school course. 1. Geometry. 1. Murrow, Gene II. Title. OA445.L36 1983 516.2 83-359 ©1983 by Springer Science+Business Media New York Originally published by Springer-Verlag New York in 1983 AII rights reserved. No part of this book may be translated or reproduced in any form without permission from Springer Science+Business Media, LLC,

9 B 7 6 5 432 ISBN 97B-0-387-90727-7 ISBN 978-1-4757-1797-6 (eBook) DOI 10.1007/978-1-4757-1797-6

Preface

A geometry course based on this book was taught successfully by Gene Murrow for several years.

We are much indebted

to Springer-Verlag for publishing Geometry, so that others can try our approach.

The publishers and we thought it

would be appropriate to issue the book first in a preliml.nary edition, on which we would welcome comments, especially from students and teachers of the high school geometry course. Such comments can bear on any aspect of Geometry, ranging from the choice of topics, the ordering of the topics, and other global considerations, to possible computational errors and misprints.

We shall welcome criticisms and

suggestions.

Serge Lang Gene Murrow

Contents

Theorems Proved in Geometry Introduction

xi xvii

CHAPTER 1 - Distance and Angles 51.

Lines

52.

Distance

12

53.

Angles

20

54.

Proofs

43

55.

Right Angles and Perpendicularity

52

86.

The Angles of a Triangle

65

1

CHAPTER 2 - Coordinates 51.

Coordinate Systems

85

52.

Distance between Points on a Line

94

53.

Equation of a Line

96

CHAPTER 3 - Area and the Pythagoras Theorem 51.

The Area of a Triangle

107

S2.

The Pythagoras Theorem

125

viii

CONTENTS

CHAPTER 4 - The Distance Formula Sl.

Distance between Arbitrary Points

142

S2.

Higher Dimensional Space

148

S3.

Equation of a Circle

155

CHAPTER 5 - Some Applications of Right Triangles S1.

Perpendicular Bisector

162

S2.

Isosceles and Equilateral Triangles

175

S3.

Theorems About Circles

190

CHAPTER 6 - Polygons S1.

Basic Ideas

205

S2.

Convexity and Angles

209

S3.

Regular Polygons

213

CHAPTER 7 - Congruent Triangles S1.

Euclid's Tests for Congruence

223

S2.

Some Applications of Congruent Triangles

239

S3.

Special Triangles

248

CHAPTER 8 - Dilations and Similarities S1.

Definition

259

S2.

Change of Area under Dilation

269

S3.

Change of Length under Dilation

287

S4.

The Circumference of a Circle

291

S5.

Similar Triangles

305

CONTENTS

ix

CHAPTER 9 - Volumes Sl.

Boxes and Cylinders

324

S2.

Cones and Pyramids

335

S3.

Change of Volume under Dilations

341

CHAPTER 10 - Vectors and Dot Product Sl.

Vector Addition

355

S2.

The Scalar Product

360

S3.

Perpendicularity

365

S4.

Projections

371

S5.

Ordinary Equation for