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A New Conception of Geometry

Jingzhong Zhang
Saltire Software
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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Here Professor Zhang shares a toolkit for solving geometry problems based on methods “used successfully by Chinese students training for the International Mathematical Olympiads.” Indeed, many examples and exercises draw on problems presented at Olympiads over the decades. These tools emerge in the proof-driven development of nine theorems with somewhat idiosyncratic names such as Co-Side Theorem, Co-Angle Converse Theorem, his often-applied Area Method, etc. Some more of the language with room for improvement in this translated work includes counterexamples being “thinking from the contrary” and line segments being “lines” while a line is an “extension line”.

The basic approach is to use fundamental knowledge around the area of the triangle, supplementary and complementary angles, and to effectively transform even complex planar representations to straightforward algebraic equations. Besides practice for competition, lecture source material abounds here from enlightening exercises to classroom capsules that could constitute the meat of a lecture. For instance, connecting trisecting points on opposing side of an irregular quadrilateral leads to the surprising result that the center of the 9-part grid has one ninth the total area. In textbooks, I have seen this economically presented on one page. Here, eight pages are devoted to generalizing the problem and exploring variations. Similar teaching value lies in the explorations of Pappus's Hexagon Theorem leading to the Newton Line.

Because of Zhang’s long history in preparing students for Olympiads, selected problems reach back into the 1970s. In some cases, the author reveals some history on the development of a problem for competition. As a result, despite the appearance of a textbook, this is more like guided preparation for student participants and coaches of mathematics competitions. Indeed, the culminating content is two chapters: “Selected Area Problems from Mathematical Competitions” and “Using the Area Method to Solve Mathematics Competition Problems.”

Tom Schulte would like to see more geometry content in undergraduate algebra courses and looks for opportunity as an instructor to bring that in.

The table of contents is not available.