Problems and Solutions in Euclidean Geometry

275 Pages·2010·3.63 MB·English

M. N. AREF AND WILLIAM WERNICK
INS
PROBLEMS &
SOLUTIONS
EUCLIDEAN
GEOMETRY
DOVER BOOKS ON MATHEMATICS
HANDBOOK OF MATHEMATICAL FUNCTIONS: WITH FORMULAS, GRAPHS, AND MATHEMATICAL
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ABSTRACT AND CONCRETE CATEGORIES: THE JOY OF CATS, Jiri Adamek, Horst Herrlich,
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NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS, Sergio
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MATHEMATICS: ITS CONTENT, METHODS AND MEANING, A. D. Aleksandrov, A. N. Kolmogo-
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COLLEGE GEOMETRY: AN INTRODUCTION TO THE MODERN GEOMETRY OF THE TRIANGLE AND THE
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THE WORKS OF ARCHIMEDES, Archimedes. Translated by Sir Thomas Heath.
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REAL VARIABLES WITH BASIC METRIC SPACE TOPOLOGY, Robert B. Ash. (0-486-47220-5)
INTRODUCTION TO DIFFERENTIABLE MANIFOLDS, Louis Auslander and Robert E. MacKenzie.
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PROBLEM SOLVING THROUGH RECREATIONAL MATHEMATICS, Bonnie Averbach and Orin
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THEORY OF LINEAR OPERATIONS, Stefan Banach. Translated by F. Jellett. (0-486-46983-2)
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ADVANCED TRIGONOMETRY, C. V. Durell and A. Robson. (0-486-43229-7)
FOURIER ANALYSIS IN SEVERAL COMPLEX VARIABLES, Leon Ehrenpreis. (0-486-44975-0)
THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 1, Euclid. Edited by Thomas L. Heath.
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THE THIRTEEN BOOKS OF THE ELEMENTS, VOL. 3, Euclid. Edited by Thomas L. Heath.
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AN INTRODUCTION TO DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS, Stanley J. Farlow.
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PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS, Stanley J. Farlow.
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STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATIONS, Avner Friedman. (0-486-45359-6)
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POINT SET TOPOLOGY, Steven A. Gaal. (0-486-47222-1)
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(continued on back flap)

PROBLEMS SOLUTIONS
AND
IN
EUCLIDEAN GEOMETRY

PROBLEMS SOLUTIONS
AND
IN
EUCLIDEAN GEOMETRY
M. N. AREF
Alexandria University
WILLIAM WERNICK
City College, NewYork
DOVER PUBLICATIONS, INC
MINEOLA, NEW YORK
Copyright
Copyright © 1968 by Dover Publications, Inc.
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2010, is a reissue of a work originally
published by Dover Publications, Inc, in 1968.
International Standard Book Number
ISBN-13: 978-0-486-47720-6
ISBN-10: 0-486-47720-7
Manufactured in the United States by Courier Corporation
47720701
www.doverpublications.com
Dedicated to my father
who has been always of lifelong inspiration
and encouragement to me
M.N.A.

CONTENTS
PREFACE xi
SYMBOLS EMPLOYED IN THIS BOOK xiii
CHAPTER 1 - Triangles and Polygons
Theorems and corollaries I
Solved problems 4
Miscellaneous exercises 25
CHAPTER 2 - Areas, Squares, and Rectangles
Theorems and corollaries 33
Solved problems 35
Miscellaneous exercises 56
CHAPTER 3 - Circles and Tangency
Theorems and corollaries 64
Solved problems 67
Miscellaneous exercises 95
CHAPTER 4 -Ratio and Proportion
Theorems and corollaries 105
Solved problems io8
Miscellaneous exercises 133
CHAPTER 5 Loci and Transversals
Definitions and theorems 144
Solved problems 145
Miscellaneous exercises 169
CHAPTER 6 -Geometry of Lines and Rays
HARMONIC RANGES AND PENCILS
Definitions and propositions 178
Solved problems 182
ISOGONAL AND SYMMEDIAN LINES-BROCARD POINTS
Definitions and propositions 186
Solved problems 189
Miscellaneous exercises 191
ix