College Geometry with GeoGebra

From two authors who embrace technology in the classroom and value the role of collaborative learning comes College Geometry Using GeoGebra, a book that is ideal for geometry courses for both mathematics and math education majors. The book's discovery-based approach guides students to explore geometric worlds through computer-based activities, enabling students to make observations, develop conjectures, and write mathematical proofs. This unique textbook helps students understand the underlying concepts of geometry while learning to use GeoGebra software—constructing various geometric figures and investigating their properties, relationships, and interactions. The text allows students to gradually build upon their knowledge as they move from fundamental concepts of circle and triangle geometry to more advanced topics such as isometries and matrices, symmetry in the plane, and hyperbolic and projective geometry.

Emphasizing active collaborative learning, the text contains numerous fully-integrated computer lab activities that visualize difficult geometric concepts and facilitate both small-group and whole-class discussions. Each chapter begins with engaging activities that draw students into the subject matter, followed by detailed discussions that solidify the student conjectures made in the activities and exercises that test comprehension of the material. Written to support students and instructors in active-learning classrooms that incorporate computer technology, College Geometry with GeoGebra is an ideal resource for geometry courses for both mathematics and math education majors.

Preface

Especially for Students

Notes for Instructors

Our Motivation, Philosophy, and Pedagogy

Prerequisites and Chapter Dependencies

Acknowledgments

ONEUsing GeoGebra

1.1 Activities: Getting Started with GeoGebra

1.2 Discussion: Exploring and Conjecturing

Some GeoGebra Tips

Constructing −→ Exploring −→ Conjecturing:

Inductive Reasoning

Language of Geometry

Explorations, Observations, Questions

The Family of Quadrilaterals

Angles Inscribed in Circles

Rules of Logic

1.3 Exercises

1.4 Chapter Overview

TWO  Constructing → Proving

 2.1 Activities

 2.2 Discussion: Euclid’s Postulates and Constructions

 Euclid’s Postulates

 Congruence and Similarity

 Constructions

 Geometric Language Revisited

 Conditional Statements: Implication

 Using Robust Constructions to Develop a Proof

 Angles and Measuring Angles

 Constructing Perpendicular and Parallel Lines

 Properties of Triangles

 Euclid’s Parallel Postulate

 Euclid’s Constructions in the Elements

 Ideas About Betweenness

 2.3 Exercises

 2.4 Chapter Overview

THREE Mathematical Arguments and Triangle Geometry

3.1 Activities

3.2 Discussion

Deductive Reasoning

Universal and Existential Quantifiers

Negating a Quantified Statement

Direct Proof and Disproof by Counterexample

Step-by-Step Proofs

Congruence Criteria for Triangles

The Converse and the Contrapositive

Concurrence Properties for Triangles

Ceva’s Theorem and Its Converse

Brief Excursion into Circle Geometry

The Circumcircle of ΔABC

The Nine-Point Circle: A First Pass

Menelaus’ Theorem and Its Converse

3.3 Exercises

3.4 Chapter Overview

FOUR Circle Geometry and Proofs

 4.1 Activities

 4.2 Discussion

 Axiom Systems: Ancient and Modern Approaches

 Language of Circles

 Inscribed Angles

 Mathematical Arguments

 Additional Methods of Proof

 Cyclic Quadrilaterals

 Incircles and Excircles

 Some Interesting Families of Circles

 The Arbelos and the Salinon

 Power of a Point

 The Radical Axis

 The Nine-Point Circle: A Second Pass

 4.3 Exercises

 4.4 Chapter Overview

FIVE Analytic Geometry

 5.1 Activities

 5.2 Discussion

 Points

 Lines

 Distance

 Using Coordinates in Proofs

 Another Look at the Radical Axis

 Polar Coordinates

 The Nine-Point Circle, Revisited

 5.3 Exercises

 5.4 Chapter Overview

SIX Taxicab Geometry

  6.1 Activities

  6.2 Discussion

  An Axiom System for Metric Geometry

  Circles

  Ellipses

  Measuring Distance from a Point to a Line

  Parabolas

  Hyperbolas

  Axiom Systems

  6.3 Exercises

  6.4 Chapter Overview

SEVEN  Finite Geometries

7.1 Activities

7.2 Discussion

An Axiom System for an Affine Plane

An Axiom System for a Projective Plane

Duality

Relating Affine Planes to Projective Planes

Coordinates for Finite Geometries

7.3 Exercises

7.4 Chapter Overview

EIGHTTransformational Geometry

 8.1 Activities

 8.2 Discussion

 Transformations

 Isometries

 Other Transformations

 Composition of Isometries

 Inverse Isometries

 Using Isometries in Proofs

 Isometries in Space

8.3 Exercises

8.4 Chapter Overview

NINE  Isometries and Matrices

9.1 Activities

9.2 Discussion

 Using Vectors to Represent Translations

 Using Matrices to Represent Rotations

 Using Matrices to Represent Reflections

 Composition of Isometries

 The General Form of a Matrix Representation

 Using Matrices in Proofs

 Similarity Transformations

 9.3 Exercises

 9.4 Chapter Overview

TENSymmetry in the Plane

 10.1 Activities

 10.2 Discussion

 Symmetries

 Groups of Symmetries

 Classifying Figures by Their Symmetries

 Friezes and Symmetry

 Wallpaper Symmetry

 Tilings

 10.3 Exercises

 10.4 Chapter Overview

ELEVEN  Hyperbolic Geometry

 Part I: Exploring a New Universe

 11.1 Activities Part I

 11.2 Discussion Part I

  Hyperbolic Lines and Segments

  The Poincaré Disk Model of the Hyperbolic Plane

  Measuring Distance in the Poincaré Disk Model

  Hyperbolic Circles

  Hyperbolic Triangles

  Circumcircles and Incircles of Hyperbolic Triangles

  Congruence of Triangles in the Hyperbolic Plane

 Part II: The Parallel Postulate in Hyperbolic Geometry

 11.3 Activities Part II

 11.4 Discussion Part II

  The Hyperbolic and Elliptic Parallel Postulates

  The Angle of Parallelism

  The Exterior Angle Theorem

  Quadrilaterals in the Hyperbolic Plane

  Another Look at Triangles in the Hyperbolic Plane

 Area in the Hyperbolic Plane

 11.5 Exercises

  The Upper-Half-Plane Model

 11.6 Chapter Overview

TWELVE Projective Geometry

12.1 Activities

12.2 Discussion

  An Axiom System

  Models for the Projective Plane

  Duality

  Coordinates for Projective Geometry

  Projective Transformations

12.3 Exercises

12.4 Chapter Overview

APPENDIX A Trigonometry

A.1 Activities

A.2 Discussion

  Right Triangle Trigonometry

  Unit Circle Trigonometry

  Solving Trigonometric Equations

  Double Angle Formulas

 Angle Sum Formulas

 Half-Angle Formulas

 The Law of Sines and the Law of Cosines

A.3 Exercises

APPENDIX B Calculating with Matrices

B.1 Activities

B.2 Discussion

  Linear Combinations of Vectors

  Dot Product of Vectors

  Multiplying a Matrix Times a Vector

  Multiplying Two Matrices

 The Determinant of a Matrix

B.3 Exercises

BIBLIOGRAPHY

INDEX

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