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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010192659 | QA564 W35 2006 | Open Access Book | Book | Searching... |
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Summary
Summary
Translated from the original German by Peter Hilton and Jean Pedersen. The 99 points of intersection presented here were collected during a year-long search for surprising concurrence of lines. For each example we find compelling evidence for the sometimes startling fact that in a geometric figure three straight lines, or sometimes circles, pass through one and the same point. Of course, we are familiar with some examples of this from basic elementary geometry - the intersection of medians, altitudes, angle bisectors, and perpendicular bisectors of sides of a triangle. Here there are many more examples - some for figures other than triangles, some where even more than three straight lines pass through a common point. The main part of the book presents 99 points of intersection purely visually. They are developed in a sequence of figures, many without caption or verbal commentary. In addition the book contains general thoughts on and examples of the points of intersection, as well as some typical methods of proving their existence. Many of the examples shown in the book were inspired by questions and suggestions made by students and high-school teachers. Several of those examples have not only a geometrical, but also an intriguing aesthetic, aspect. The book addresses high-school students and students at the undergraduate level as well as their teachers, but will appeal to anyone interested in geometry.
Reviews 1
Choice Review
The heart of this book consists of 99 diagrams that illustrate, without any verbal explanation, 99 theorems of plane geometry. In each case, the theorem would assert that three (or more) lines (or occasionally circles or ellipses) all pass through one common point. A short introduction, mostly independent from the 99 examples, establishes the general theme, and 30 pages of more or less traditionally presented mathematical background round out the volume. In the concluding material, one finds proofs for just a few of the 99 theorems. Teachers will find those 99 pictures usual as challenging exercises for a traditional Euclidean geometry course, or alternatively as opportunities to demonstrate tools of computer algebra. As Walser (ETH Zurich; Univ. of Basel) notes, existing "dynamic geometry software" can animate such examples. Still shy of proof, animations almost remove all of a viewer's doubt; they may also foster additional geometrical insight. Even accepting the value of setting out these theorems without proof, verbal statements of the theorems would help. As it stands, readers must sometimes guess when the author means to take a certain triangle as equilateral or some pair of triangles as similar, etc. ^BSumming Up: Optional. General readers; lower-division undergraduates; faculty. D. V. Feldman University of New Hampshire
Table of Contents
| Part I What's It All About? |
| 1 If three lines meet |
| 2 Flowers for Fourier |
| 3 Chebyshev and the Spirits |
| 4 Sheaves generate curves |
| Part II The 99 points of intersection: |
| Part III The Background |
| 1 The four classical points of intersection |
| 2 Proof strategies |
| 3 Central projection |
| 4 Ceva's Theorem |
| 5 Jacobi's Theorem |
| 6 Remarks on selected points of intersection |
| References |
