Clayton W. Dodge - Euclidean Geometry and Transformations

PREFACE

Just as analytic geometry is recognized today as an important tool in geometry, so also are isometries and similarities important geometric tools. It is well known that Euclidean geometry is the study of those properties of points that are invariant under isometries and similarities, but just how such properties are exhibited using these transformations has not been widely discussed in textbooks. A primary purpose of this book is to provide a source for both the theory and the practical application to geometry of these transformations for college students of mathematics in general, and for teachers and prospective teachers of geometry in particular.

The spirit of modern elementary geometry is also presented with topics such as Menelaus and Cevas theorems, Euclidean constructions, and the geometry of special lines and points associated with a triangle, thereby reviewing and refreshing the readers memory for high school geometry and preparing him to dogeometry. The high school geometry teacher who has mastered this text can be confident that he is prepared to handle the geometry problems that arise in high school classes.

Prerequisites for this material include high school algebra, geometry, and elementary trigonometry. In addition, some familiarity with the concept of function will prove helpful.

The primary goal of this book is to prepare the reader to do Euclidean geometry. Hence much of it is written in the style of the classic College Geometryby N. A. Court. The reader is given many opportunities to work exercises, for such is the key to understanding mathematics. It is suggested that the reader pause a moment after reading the statement of each theorem in the text, draw an appropriate figure, and attempt a proof of the theorem before reading further. Compare the attempted proof with the proof given in the text. Work an abundance of exercises. Look first in the section of Hints when unable to obtain a solution, then look at the Answers section only as a last resort. Steady progress toward genuine understanding will result.

Geometry, when understood, is indeed a fascinating study.

Each chapter begins with a section of history or commentary which need not be assigned for formal class study. Although exercises are provided for these sections, their purposes are to whet the appetite of the student and to provide some enrichment material.

Following the commentary section in each of , one or two sections entitled Introduction.. inform the reader of the theory that is developed in the sections that follow. The casual reader may wish to skip over the formal development at that time and look directly at the Applications sections, returning later to fill in theoretic gaps.