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
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