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Methods of Solving Complex Geometry Problems

Ellina Grigorieva
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Mark Hunacek
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The word “complex” appearing in the title of this book modifies “problems”, not “geometry”, so a reader looking for a collection of problems in algebraic geometry over the field of complex numbers will have to look elsewhere. What we have here is an interesting and quite well-chosen collection of problems in Euclidean plane geometry. The problems are elementary in the sense that the geometry that is covered here would likely be comprehensible to a high-school student (although there are topics covered in this book that would not typically be taught in high school, such as the theorems of Ceva and Menelaus) but some of them are very challenging.

The book is divided into four chapters, covering, respectively, triangles, quadrilaterals, circles, and geometric constructions. Each chapter is divided into sections, and frequently subsections, consisting of statements (and usually proofs) of standard results, followed by numbered problems with solutions appearing immediately after the statement. (There are a total of 81 of these.) In addition to these numbered problems, the final section of each chapter consists of 20 problems labeled “homework”; these are generally accompanied by solutions, some more detailed than others. The methods used are those of elementary synthetic geometry and trigonometry, but Menelaus’ theorem is also given a vector formulation and on one or two occasions the geometric-transformation approach is exploited. The problems range in difficulty from the obvious to the quite difficult; some call for proofs, but quite a few call for computations or constructions.

Considerations of space obviously make it impracticable to list all or even many of the problems here, but the following verbatim selection of four problems, one from each chapter, should give a reasonable indication of what they look like:

· Consider a scalene triangle ABC with area S. Point P is in the interior of ABC. Draw three lines through P parallel to each side of the triangle to form three triangles with areas S1, S2 and S3. Find S.

· A parallelogram with area S is given. Each vertex of the parallelogram is connected by a line segment to the midpoints of the two opposite sides. Find the area of the polygon that is formed by the intersection of the all [sic] line segments.

· Let AB be a diameter of a circle and let CD be a chord. The chord is not perpendicular to the diameter. Prove that the perpendiculars AE and BF dropped from the end of the diameter to the chord cuts [sic] the chord CD into equal segments CF and DE.

· Construct a triangle by its two vertices, A and B, and its orthocenter, H.

These seemed to me to be some of the more challenging problems. Others, as I have indicated, are much easier, even trivial: e.g., “Can you construct an angle of 30 There are also some that, although not obvious, are fairly well known: “Prove that if consecutive midpoints of all sides of a quadrilateral are connected, that they form a parallelogram.”

The author — who, according to the preface, initially disliked and was not all that good at geometry, but grew to love it — has made a special effort to make this book as painless as possible for the student. In fact, in what is clearly an effort to “humanize” things, she has created a small Appendix of four photos (three in color), and also reproduced a page from a notebook she kept in the ninth grade. One certainly doesn’t see things like this in very many textbooks.

In connection with the above, it sometimes appeared to me that the author made too much of an effort to make things painless; one or two problems, for example, are stated in a way that I thought too cloyingly juvenile for a Birkhäuser text. (Problem 28, for example, reads: “Peter has drawn a triangle with sides 11, 13 and 6. Can you help him to evaluate the area of the triangle?”).

In addition, the book would have benefited from more careful editing. I noticed several typos (such as the ones reprinted above, as well as the first paragraph on page 45, which ends mid-sentence, and the caption to figure 3.15, which reads: ”The angle between two secants (must be replaced).” There is also at least one substantive mathematical issue, where the author purports (on page 147) to give a proof of the fact that a quadrilateral can be inscribed in a circle if and only if the sum of the opposite angles is 180 degrees, but actually proves only the “only if” direction.

Fortunately, however, these issues do not seriously detract from the overall value of this book. While not intended as a text, it nevertheless is versatile enough to serve several purposes and different constituencies. It gives a good overview of a number of different topics in Euclidean geometry, and provides excellent practice for people preparing to engage in Olympiad-style competitions. It also makes for nice supplementary reading for students (or more likely teachers) of geometry. I am currently teaching a course in Euclidean geometry, and I doubt it will be very long before I turn to this book for interesting supplemental homework problems.

Mark Hunacek ( teaches mathematics at Iowa State University.

1 Problems Involving Triangles

2 Quadrilaterals and other Polygons

3 Problems Involving Circles

4 Problems on Construction

Appendix A Ratios and Proportion

Appendix B My 9th Grade Notebook Page

Appendix C My Pictures