Geometric Transformations

✍ Scribed by Răzvan GelcaIonuţ OnişorCarlos Yuzo Shine

• Publisher: Springer, Cham
• Year: 2022
• Tongue: English
• Leaves: 581
• Series: Problem Books in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on some classical examples and theorems. It is based on the combined teaching experience of the authors (coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations. The book offers a global view of the geometric content of today's mathematics competitions, bringing many new methods and ideas to the attention of the public.

Talented high school and middle school students seeking to improve their problem-solving skills can benefit from this book, as well as high school and college instructors who want to add nonstandard questions to their courses. People who enjoy solving elementary math problems as a hobby will also enjoy this work.

✦ Table of Contents

Preface
Contents
Part I Problems
1 Isometries
1.1 Theoretical Results About Isometries
1.1.1 Definition and Basic Properties
1.1.2 Translations, Rotations, Reflections
1.1.3 Isometries as Composition of Reflections
1.1.4 Compositions of Isometries
1.1.5 Discrete Groups of Isometries
1.1.6 Theoretical Questions About Isometries
1.2 Isometries in Euclidean Geometry Problems
1.2.1 Some Constructions and Classical Results in Euclidean Geometry That Use Isometries
1.2.2 Examples of Problems Solved Using Isometries
1.2.3 Problems in Euclidean Geometry to be Solved Using Isometries
1.3 Isometries Throughout Mathematics
1.3.1 Geometry with Combinatorial Flavor
1.3.2 Combinatorics of Sets
1.3.3 Number Theory
1.3.4 Functions
2 Homotheties and Spiral Similarities
2.1 A Theoretical Introduction to Homotheties
2.1.1 Definition and Properties
2.1.2 Groups Generated by Homotheties
2.1.3 Problems About Properties of Homotheties
2.2 Problems in Euclidean Geometry That Use Homothety
2.2.1 Theorems in Euclidean Geometry Proved Using Homothety
2.2.2 Examples of Problems Solved Using Homothety
2.2.3 Problems in Euclidean Geometry to be Solved Using Homothety
2.3 Homothety in Combinatorial Geometry; Scaling
2.4 A Theoretical Study of Spiral Similarities
2.4.1 The Definition and Properties of Spiral Similarities
2.4.2 The Center of a Spiral Similarity: The Generic Case
2.4.3 The Center of a Spiral Similarity: The Case A'=B, Symmedians Revisited
2.4.4 Spiral Similarities and Miquel's Theorem
2.4.5 Compositions of Spiral Similarities
2.4.6 Groups Generated by Spiral Similarities
2.4.7 Theoretical Questions About Spiral Similarities
2.5 Spiral Similarity in Euclidean Geometry Problems
2.5.1 Similar Figures and the Circle of Similitude
2.5.2 Examples of Problems Solved Using Spiral Similarities
2.5.3 Problems in Euclidean Geometry to be Solved Using Spiral Similarities
3 Inversions
3.1 Theoretical Results About Inversion
3.1.1 The Definition of Inversion and Some of Its Properties
3.1.2 Inverses of Lines and Circles
3.1.3 Möbius Transformations
3.1.4 Möbius Transformations Versus Isometries, Spiral Similarities, and Inversions; Inversion and Circular Transformations
3.1.5 Linear Fractional Transformations of the Real Line
3.1.6 The Invariance of Angles
3.1.7 Inversion with Negative Ratio
3.1.8 Circles Orthogonal to the Circle of Inversion
3.1.9 The Limiting Points of Two Circles
3.1.10 Problems with Theoretical Flavor About Properties of Inversion and Möbius Transformations
3.2 Inversion in Euclidean Geometry Problems
3.2.1 Applications of Inversion to Proving Classical Results
3.2.2 Examples of Problems Solved Using Inversion
3.2.3 Problems in Euclidean Geometry to be Solved with Inversion (or with Möbius Transformations)
4 A Synthesis
4.1 Bringing Together All Transformations
4.1.1 Some Examples
4.1.2 Some Problems
4.2 A Story of Complete Quadrilaterals
4.2.1 Miquel's Theorem and bc Inversion
4.2.2 Some Classical Results
4.2.3 Problems About Complete Quadrilaterals
Part II Hints
5 Isometries
6 Homotheties and Spiral Similarities
7 Inversions
8 A Synthesis
Part III Solutions
9 Isometries
10 Homotheties and Spiral Similarities
11 Inversions
12 A Synthesis
Index