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The Circle: A Mathematical Exploration Beyond the Line
Author(s): Alfred S. Posamentier (Author), Robert Geretschläger (Author)
- Publisher: Prometheus Books
- Publication Date: 23 Aug. 2016
- Language: English
- Print length: 300 pages
- ISBN-10: 1633881679
- ISBN-13: 9781633881679
Book Description
The authors begin with a brief review of the basic properties of the circle and related figures. They then show the many ways in which the circle manifests itself in the field of geometry--leading to some amazing relationships and truly important geometric theorems. In addition, they explore remarkable circle constructions and demonstrate how all constructions in geometry that usually require an unmarked straightedge and a compass can also be done with the compass alone. Among other things, the reader will learn that circles can generate some unusual curves - many even quite artistic. Finally, the role of circles in art and architecture and a discussion of the circle's place on the sphere bring "full circle" this presentation of a key element of geometry.
Editorial Reviews
Review
—Arthur Mettinger, professor, University of Vienna, Department of English and American Studies; and academic president, FH Campus Wien University of Applied Sciences
“This book should be read by everyone who thinks math and geometry are difficult or boring. The authors, leading mathematicians themselves, have taken the topic of circles and shown us how fascinating—magic, indeed—and central to our experience of the world circles are. Posamentier and Geretschläger have explored the world of circles and their many applications in everyday life as well as in geometry, in a way that the most nervous layperson can understand and enjoy. I for one will never look at architecture or art in the same way after reading this book. I especially commend it to every teacher or confused student of mathematics.”
—Baroness Pauline Perry, former president of Lucy Cavendish College, Cambridge University
“After reading this wonderful book, which helped me to recall my early memories of the geometry of circles, I was absolutely enchanted by the wonderful relationships involving circles—as well as their constructions and their appearances in our culture and on the sphere—that we should have been exposed to in our school days but never were! This is the book that all who have even the slightest interest in geometry will love, to have their knowledge of circles enriched. Great book!”
—Charlotte Frank, PhD, senior advisor, McGraw-Hill Education; former director of curriculum and instruction, New York City Board of Education; and former regent, University of the State of New York
“This is a remarkable addition to the geometry literature, giving a very incisive new insight into the role of the circle and how it relates to the line and other geometrical forms. Whatever your background, this is sure to help you reinforce your comprehension of all things geometric.”
—Peter Taylor (AO), emeritus professor, University of Canberra
“Alfred S. Posamentier and Robert Geretschläger are a congenial mathematical author team; they are impressive not only for their superior content competence but also for their extraordinary talent to present mathematics to the general audience as a fascinating, enriching adventure. This book not only enhances the reader’s knowledge but also is remarkable for its modest tone and splendid style.”
—Rudolf Taschner, professor of mathematics, Vienna University of Technology, and 2004 Austrian Scientist of the Year
“This volume is a treasure trove on everything you want to know about circles. Starting from simple results in Euclidean geometry, it reaches far both in space and time, revisiting many famous historical problems and considering more modern topics such as inversive geometry and combinatorial geometry. . . . A fantastic collaboration between two noted mathematicians.”
—Andy Liu, Professor Emeritus, Department of Mathematical and Statistical Sciences, University of Alberta, and vice president, International Mathematics Tournament of the Towns
About the Author
Excerpt. © Reprinted by permission. All rights reserved.
The Circle
A Mathematical Exploration beyond the Line
By Alfred S. Posamentier, Robert Geretschläger
Prometheus Books
Copyright © 2016 Alfred S. Posamentier and Robert Geretschläger
All rights reserved.
ISBN: 978-1-63388-167-9
Contents
Acknowledgments, 9,
Introduction, 11,
1. Circle Relationships in Basic Plane Geometry — and an Extension, 15,
2. The Circle's Special Role in Geometry, 33,
3. Famous Theorems about Circles, 81,
4. Circle-Packing Problems, 113,
5. Investigation of Equicircles, 139,
6. Circle Constructions: The Problem of Apollonius, 155,
7. Inversion: Circle Symmetry, 175,
8. Mascheroni Constructions: Using Only Compasses, 197,
9. Circles in Art and Architecture, 217,
10. Rolling Circles: Hypocycloids and Epicycloids by Christian Spreitzer, 233,
11. Spherical Geometry — Circles on the Sphere, 261,
Afterword: A Cultural Introduction to the Circle by Erwin Rauscher, 281,
Appendixes, 297,
Notes, 323,
Bibliography, 329,
Index, 331,
CHAPTER 1
CIRCLE RELATIONSHIPS IN BASIC PLANE GEOMETRY — AND AN EXTENSION
As we begin our exploration of the circle, the one aspect that most people recall is that the Greek letter p (pi) is somehow related to this important geometric shape. Most will recall that p represents the ratio of the circumference of a circle to its diameter. The two formulas that come instantly to mind are that the circumference of the circle is equal to 2pr, and that the area of the circle is equal to pr. In each case, r is the length of the radius of the circle. However, before we inspect the circle for its multitude of applications and appearances, we will review some of the basic essentials that you may have forgotten from your high-school geometry course.
First, we will review some basic terminology related to circles. We all know that the radius is the line segment joining the center of the circle to any point on the circle, while the diameter is the line segment joining two points on the circle and that also contains the center of the circle. A line segment joining two points on the circle is called a chord. A line is said to be tangent to the circle if it touches a circle at exactly one point; and a line that intersects the circle in two points is called a secant. Two more definitions to review: the area within a circle bounded by two radii and the arc between them is called a sector, and the area within a circle bounded by a chord and the arc of the circle is called a segment of the circle.
Some relationships useful to our study of the circle are the following:
• Any three non-collinear points determine a unique circle.
* The non-collinear points A, B, and C determine the unique circle O. (See figure 1.1.)
• The perpendicular bisector of a chord contains the center circle and the midpoints of the arcs it intersects.
* Line CD is the perpendicular bisector of AB; therefore, line CD contains the center of the circle and points C and D are the midpoints of arcs AB. (See figure 1.2.)
• A line perpendicular to a radius at the endpoint on the circle is tangent to the circle.
* Line AB is perpendicular to radius OC at point C and is therefore tangent to the circle at point C. (See figure 1.3.)
• From a point outside the circle, two tangents drawn to the circle have equal segments to the points of tendency.
* Tangents PA and PB are equal in length. (See figure 1.4.)
• A polygon is said to be inscribed in a circle if all of its vertices are on the circle.
* Polygon ABCDE is inscribed in circle O. (See figure 1.5.)
• A circle is said to be inscribed in a polygon if all of the polygon's sides are tangent to the circle. (See figure 1.6.)
* Circle O is inscribed in polygon ABCDE.
• If a secant segment and a tangent segment to the same circle share an endpoint in the exterior of the circle, then the length of the tangent segment is the mean proportional between the length of the secant segment and the length of its external segment.
* Tangent AP is the mean proportional between PC and PB, so that
PC/AP = AP/PB
(see figure 1.7).
• If two secant segments of the same circle share an endpoint in the exterior of the circle, then the product of the lengths of one secant segment and its external segment equals a product of the lengths of the other secant segment and its external segment.
* For the two secants PED and PBC, the following is true:
PD · PE = PC · PB. (See figure 1.8.)
• If two chords intersect in the interior of a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
* For the two chords intersecting at the point P, the following is true: PA · PB = PC · PD. (See figure 1.9.)
• A central angle is one formed by two radii of a circle, and it is equal in measure to its intercepted arc.
* The measure of angle AOB is equal to the degree measure of arc AB. (See figure 1.10.)
• The measure of an inscribed angle, one formed by two chords intersecting on the circle, is one half the measure of its intercepted arc.
* The measure of angle APB is one-half the measure of arc AB. (See figure 1.11.)
• The measure of an angle formed by a tangent and a chord of a circle is one half the measure of its intercepted arc.
* The measure of angle ABP is one-half the measure of arc AB. (See figure 1.12.)
• The measure of an angle formed by two chords intersecting at a point in the interior of the circle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
* The measure of angle BPD is equal to one-half the measure of the sum of the measures of arcs BD and AC. (See figure 1.9.)
• The measure of an angle formed by two secants of the circle intersecting at a point in the exterior of the circle is equal to one-half the difference of the measures of the intercepted arcs.
* The measure of angle DPC is one-half the measure of arc DC minus arc EB. (See figure 1.8.)
• The measure of an angle formed by a secant and a tangent to a circle, which intersect at a point in the exterior of the circle, is equal to one-half the difference of the measures of the intercepted arcs.
* The measure of angle APC is equal to one-half the measure of arc AC minus arc AB. (See figure 1.7.)
• The measure an angle formed by two tangents from a common external point to a circle is equal to one half the difference of the measures of the intercepted arcs. We also notice that the measure of the angle formed by two tangents from an external point is supplementary to the measure of the closer intercepted arc.
* The measure of angle APB is equal to the difference of the two arcs AB. Additionally, the measure of angle APB is supplementary to the closer arc AB. (See figure 1.4.)
This brief review of the essentials of circle relationships that are introduced in the high-school geometry course should...
