Overview

This book provides an enquiry-based introduction to advanced Euclidean geometry. It utilises the dynamic geometry program GeoGebra to explore many of the most interesting theorems in the subject. Topics covered include triangle centres, inscribed, circumscribed and escribed circles, medial and orthic triangles, the nine-point circle, the theorems of Ceva and Menelaus, and many applications. 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 or as a stand-alone introduction to advanced topics in Euclidean geometry. The exposition consists almost entirely of exercises (with hints) that guide students as they discover the geometric relationships for themselves. The ideas are first explored at the computer and then assembled into a proof of the result under investigation, allowing readers to experience the joy of discovery and develop a deeper appreciation for the subject.

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9780883857847

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Reviews

Theorems in advanced Euclidean geometry are gifts to us all of beauty, elegance, grace, and pure pleasure. In his book Exploring Advanced Euclidean Geometry with GeoGebra, Gerard Venema presents a treasure trove of mathematical joys that we can all delight in. Every mathematically inclined person enjoys the many 'aha' moments that lie at the heart of elegant insights and proofs of geometry. One of the great insights in mathematics is how much richness even apparently simple ideas and objects contain. The apparently simple triangle and circle contain amazing structure that we are led to discover using the twin tools of thought and the GeoGebra software. An abundance of triangle centers, the Euler line, the Nine Point Circle, many other insights into triangles and circles, and much more geometry fill this book with beautiful theorems and visual insights that the reader discovers and experiences through Venema's guidance. Geometrical thinking connects us to a mathematical theme that has enlivened thought for thousands of years, Venema's book lets us experience the exploration and discovery of geometrical insights that have brought joy to human beings for millennia and will bring delights to all of us for years to come. If you want to teach a course that conveys the joy of mathematical thinking to students, consider leading your students on a journey through Venema's book. The theorems are classics and Venema's presentation allows the reader to experience the discovery of insights, which make them all the more meaningful and memorable. An Inquiry Based Learning course based on Venema's book would be a perfect experience for any mathematics major, particularly, a person who might teach geometry later. Exploring Advanced Euclidean Geometry with GeoGebra is also a perfect book for individual or small group study. Students, teachers, or anyone who enjoys geometrical beauty can take their time and savor the many wonderful theorems that this book contains. Whether used in a classroom setting or for individual instruction, every reader of Gerard Venema's Exploring Advanced Euclidean Geometry with GeoGebra is in for a feast of delectable geometry. - Michael Starbird Discovery learning (or inquiry-based learning, or Moore Method, or many other related variants) deemphasizes lecture and reading in favor of allowing students to develop on their own 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 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. Exploring Advanced Euclidean Geometry with GeoGebra is written for an inquiry-based approach, with lots of exercises and just enough narrative and historical commentary to hold it all together. It is not the sort of book you read without some paper and probably a computer in front of you. What makes the book special is the inclusion of GeoGebra exercises (clearly identified with a *) to encourage experimentation. Exercises may ask students to construct a visualization of a theorem, verify results, and build examples and conjectures. Eventually the student gets to proving a theorem, but not before playing with the statement quite a bit. The focuses on advanced planar Euclidean geometry, which the author defines to mean anything developed after Euclid's Elements. This makes it an excellent candidate text for a second course in Euclidean geometry using inquiry-based methods that minimize lecture and maximize student discovery. There is also much value to be mined as a supplement to other Euclidean geometry texts. The author suggests a structure in which this text is used as something of a lab manual rather than a primary text. Even if it does not fit for course adoption, this book is worth any geometry teacher's attention. The problem sequences shoe the author has thought much, and not just about how to guide the reader to theorem discovery. Teachers will find unfamiliar results to rediscover for themselves and perhaps reconnect with the learning experience they want for their students. As to content and prerequisites, the reader is expected to be familiar with elementary Euclidean geometry and generally be comfortable with definitions and proof. Some key Euclidean results are quickly reviewed in the first chapter. Topics include various notions of triangle center, the nine-point circle, the theorems of Ceva and Menelaus, geometric inversion, and hyperbolic geometry. It has a lot of content for a slim volume (130 pages), but that's the point - inquiry-based texts are necessarily short. There are only a couple of short chapters about GeoGebra itself, which was a good move. Print guides to software do not age well and such material is better consumed online. Venema keeps these discussions about building the tools relevant to the book's exercises and less about click here to do that. This helps make the book adaptable to other software. Advanced Euclidean geometry is an uncommon course offering these days. This book would work as a sourcebook for directed study and as such is worth consideration for many academic libraries. Inquiry-minded instructors should absolutely give this some attention. Working through these exercises feels very much like the process of doing mathematics, which is about the most one can ask of a book like this. The engaged student will learn much about learning as well as geometry. - Bill Wood, MAA Reviews

Author Information

Gerard A. Venema is a Professor of Mathematics at Calvin College, Michigan. He is a member of the American Mathematical Society and the Mathematical Association of America. In addition, he served for ten years as an Associate Editor of 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.

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