This book provides an inquiry-based introduction to advanced Euclidean geometry. It utilizes dynamic geometry software, specifically GeoGebra, to explore the statements and proofs of many of the most interesting theorems in the subject. Topics covered include triangle centers, inscribed, circumscribed, and escribed circles, medial and orthic triangles, the nine-point circle, duality, and the theorems of Ceva and Menelaus, as well as numerous applications of those theorems. The final chapter explores constructions in the Poincaré disk model for hyperbolic geometry.
The book can be used either as a computer laboratory manual to supplement an undergraduate course in geometry or as a stand-alone introduction to advanced topics in Euclidean geometry. The text consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. First the ideas are explored at the computer and then those ideas are assembled into a proof of the result under investigation. The goals are for the reader to experience the joy of discovering geometric relationships, to develop a deeper understanding of geometry, and to encourage an appreciation for the beauty of Euclidean geometry.
Interview with Gerard Venema about his Book (mp3)
Table of Contents
Quick Review of Elementary Euclidean Geometry
1. The Elements of GeoGebra
2. The Classical Triangle Centers
3. Advanced Techniques in GeoGebra
4. Circumscribed, Inscribed, and Escribed Circles
5. The Medial and Orthic
7. The Nine-Point Circle
8. Ceva's Theorem
9. The Theorem of Menelaus
10. Circles and Lines
11. Applications of the Theorem of Menelaus
12. Additional Topics in Triangle Geometry
12.1. Napoleon's theorem and the Napoleon point
13. Inversions in Circles
14. The Poincare Disk
About the Author
Gerard Venema earned an A.B. in mathematics from Calvin College and a Ph.D. from the University of Utah. After completing his education he spent two years in a postdoctoral position at the University of Texas at Austin and another two years as a Member of the Institute for Advanced Study in Princeton, NJ. He then returned to his alma mater, Calvin College, and has been a faculty member there ever since. While on the Calvin College faculty he has also held visiting faculty positions at the University of Tennessee, the University of Michigan, and Michigan State University. He spent two years as a Program Director in the Division of Mathematical Sciences at the National Science Foundation.
Venema is a member of the American Mathematical Society and the Mathematical Association of America. He served for ten years as an Associate Editor of The American Mathematical Monthly and currently sits on the editorial board of MAA FOCUS. Venema has served the Michigan Section of the MAA as chair and is the 2013 recipient of the section’s distinguished service award. He currently holds the position of MAA Associate Secretary and is a member of the Association’s Board of Governors.
Venema is the author of two other books. One is an undergraduate textbook, Foundations of Geometry, published by Pearson Education, Inc., which is now in its second edition. The other is a research monograph coauthored by Robert J. Daverman. It is titled Embeddings in Manifolds and was published by the American Mathematical Society as volume 106 in its Graduate Studies in Mathematics series. In addition to these books, Venema is author of over 30 research articles in geometric topology.
Discovery learning (or inquiry-based learning, or Moore method, or many other related variants) de-emphasizes lecture and reading in favor of allowing students to develop on their own as much of the material as possible. Euclidean geometry is an excellent playground for this because you can start with a few comprehensible common notions and postulates and run with them.
One step that is sometimes missing from the discovery learning process is the computer. Really understanding a result requires not just proof but probing deeply into the assumptions and finding examples that illustrate what is happening. In geometry in particular, there is software available to help students find the examples that lead to understanding, proofs, and new conjectures. Gerard A. Venema’s Exploring Advanced Euclidean Geometry with GeoGebra is a discovery learning text that embraces this approach. Continued...