10  Polygons with Circles pp. 117130By Claudi Alsina and Roger Nelsen
Icons of Mathematics
Dolciani Mathematical Expositions
(No. 45)
Online ISBN: 9780883859865
Chapter DOI: http://dx.doi.org/10.5948/UPO9780883859865.012
Subjects: Geometry and topology 

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“… I've something more important here. Something more lasting, you know. Something which will really endure”
“What's that?”
“Mind! Don't spoil my circles! It's the method of calculating the area of a segment of a circle.”
Karel ČapekThe Death of Archimedes
Euclid devoted Book IV of the Elements to propositions concerning inscribing and circumscribing polygons in or about circles, and circles in or about polygons. They have had a profound impact on geometry. For example,Archimedes, in Measurement of the Circle, was able to show that π is approximately 22/7 by inscribing and circumscribing regular polygons with 96 sides and computing the ratios of the perimeters of polygons to their diameters.
In Leonardo da Vinci's Vitruvian Man, seen in Figure 10.1 from his 15th century drawing and on a one euro coin from Italy, the artist is comparing the proportions of the human body to a circumscribed square and circle.
Circles with polygons—especially squares—are commonmotifs in art and everyday objects. In Figure 10.2 we see Wassily Kandinsky's 1913 painting Color Study: Squares with Concentric Rings, and similar designs on a shower curtain and a rug, which also appear in ceramics, jewelry, and corporate logos.
Every triangle has an inscribed circle and a circumscribed circle, and we discussed their properties in Chapters 6 and 7. So we begin our explorations in this chapter with quadrilaterals—cyclic, tangential, and bicentric. For cyclic quadrilaterals we present Ptolemy's theorem, discuss the anticenter, and prove the Japanese theorem, a lovely result from the sangaku tradition. We also present Fuss's theorem for bicentric quadrilaterals and the butterfly theorem for a selfintersecting quadrilateral inscribed in a circle.
Frontmatter:  pp. iviii 
Preface:  pp. ixx 
Twenty Key Icons of Mathematics:  pp. xixii 
Contents:  pp. xiiixviii 
1  The Bride's Chair:  pp. 114 
2  Zhou Bi Suan Jing:  pp. 1520 
3  Garfield's Trapezoid:  pp. 2128 
4  The Semicircle:  pp. 2944 
5  Similar Figures:  pp. 4560 
6  Cevians:  pp. 6176 
7  The Right Triangle:  pp. 7790 
8  Napoleon's Triangles:  pp. 91102 
9  Arcs and Angles:  pp. 103116 
10  Polygons with Circles:  pp. 117130 
11  Two Circles:  pp. 131148 
12  Venn Diagrams:  pp. 149162 
13  Overlapping Figures:  pp. 163172 
14  Yin and Yang:  pp. 173182 
15  Polygonal Lines:  pp. 183200 
16  Star Polygons:  pp. 201220 
17  Selfsimilar Figures:  pp. 221232 
18  Tatami:  pp. 233242 
19  The Rectangular Hyperbola:  pp. 243252 
20  Tiling:  pp. 253260 
Solutions to the Challenges:  pp. 261308 
References:  pp. 309320 
Index:  pp. 321326 
About the Authors:  pp. 327328 