Euclidean and Transformational Geometry: A Deductive Inquiry

The author examines various strategies and heuristics for approaching a proof and helps students determine how to proceed from one step to the next.

The text emphasizes strategies and heuristics of problem solving, discussing how students will know where to begin and how to proceed, which approach is more promising and why, and whether there are different possible solutions to a given problem.

Provides an in-depth exploration of planar Euclidean geometry, with many theorems and problems approached in various ways.

Includes a large collection of problems at various levels of difficulty.

Provides an in-depth discussion of constructions, with most discussed in three stages:  (i)  Investigation- where the discovery of how to construct the required figures is explored so that students can do new constructions on their own, (ii) Description of the construction steps and the actual construction, and (iii)  Proof of the construction.

0     Prologue
       1.  The Treasure Island Problem
       2.  The Nine-Point Circle
       3.  Morley's Theorem
       4.  The Hiker's Path
       5.  The Shortest Highway
       6.  Steiner's Minimum Distance Problem
       7.  The Pythagorean Theorem

1.     Congruence, Constructions, and the Parallel Postulate
       1-1   Angles and Their Measurement
       1-2   Congruences of Triangles
       1-3   The Parallel Postulate and Its Consequences
       1-4   More on Construction

2.     Circles
       2-1   Basic Properties of Arcs, Central and Inscribed Angles
       2-2   Circles Inscribed in Polygons
       2-3   More on Constructions

3.     Area and the Pythagorean Theorem
       3-1   Areas of Polygons
       3-2   The Pythagorean Theorem
       3-3   The Distance Formula

4.     Similarity
       4-1   Ratio, Proportion and Similar Polygons
       4-2   Further Applications of the Side Splitting Theorem and Similarity
       4-3   Areas of Similar Figures
       4-4   The Golden Ratio and the Construction of a Regular Pentagon
       4-5   Circumference and Area of a Circle
       4-6   Other Recursive Formulas for Evaluating p
       4-7   Trigonometric Functions

5.     Isometries
       5-1   Reflections, Translations, and Rotations
       5-2   Congruence and Euclidean Constructions
       5-3   More on Extremal Problems
       5-4   Similarity Transformation with Applications to Constructions

6.     Composition of Transformations and Transformation Groups
       6-1   In Search for New Isometries
       6-2   Composition of Rotations, The Treasure Island Problems and Other Treasures

7.     More Recent Discoveries
       7-1   The Nine-Point Circle and Other Results
       7-2   Complex Numbers and Geometry

• This book’s major strength is its clever combination of challenge, clarity, and instruction to teach ideas. The concepts are so clearly presented that students can easily learn them and the skillfully done illustrations augment the book’s clarity. Each step of instruction is included and labeled so that the student will not miss some crucial step in their thinking. Multiple paths to the solution of a problem are presented so that the student learns alternative ways of thinking about that concept. This variation and the easy to understand style of writing makes this book interesting and an intellectually stimulating read.
  

  I highly recommend “Euclidean and Transformational Geometry” to all math instructors at the middle school, high school and college levels. Not only has reading it and doing the problems myself greatly enhanced my own understanding of geometry, it has made the subject become beautifully alive for me and the students I share it with. It is a book I cannot imagine being without.

  Katie Wilkinson
  Math Teacher, Southridge High School

Ideal for college level Euclidean geometry students at the junior or senior level.

• Non-Euclidean Geometry Topics
• Non-Euclidean Geometry Topics
• Libeskind Errata
• 3-D Supplement