The authors present twenty icons of mathematics, that is, geometrical shapes such as the right triangle, the Venn diagram, and the yang and yin symbol and explore mathematical results associated with them. As with their previous books (*Charming Proofs*, *When Less is More*, *Math Made Visual*) proofs are visual whenever possible.

The results require no more than high-school mathematics to appreciate and many of them will be new even to experienced readers. Besides theorems and proofs, the book contains many illustrations and it gives connections of the icons to the world outside of mathematics. There are also problems at the end of each chapter, with solutions provided in an appendix.

The book could be used by students in courses in problem solving, mathematical reasoning, or mathematics for the liberal arts. It could also be read with pleasure by professional mathematicians, as it was by the members of the Dolciani editorial board, who unanimously recommend its publication.

4.2. Label the vertices of the triangle as shown in Figure 4.2b, and let O denote the center of the semicircle. Then jAOj D jBOj D jCOj since each is the length of a radius, and thus triangles AOC and BOC are isosceles. Hence †OAC D †OCA D ˛ and †OBC D †OCB D ˇ. The angles of triangle ABC sum to 180ı , so 2˛ C 2ˇ D 180ı , or ˛ C ˇ D 90ı . Thus C is a right angle as claimed. Also, 2˛ C 2ˇ D 180ı implies that †BOC D 2˛. Thales’ triangle theorem is used repeatedly throughout this chapter. The

the right triangle ABC indicated by the dashed lines in Figure 7.14b are parallel to one another and perpendicular to the line of the hypotenuse of ABC . (a) (b) Ic B Ia A C Ib Figure 7.14. 7.4. In [Williamson, 1953] we find a rule for generating right triangles with rational sides: Take any two rational numbers whose product is 2 and add 2 to each. The results are the legs of right triangle with rational sides. For example, .7=3/.6=7/ D 2, so 13=3 and 20=7 are sides of right triangle with a

C : KD 2 3 ✐ ✐ ✐ ✐ ✐ ✐ “MABK018-08” — 2011/5/16 — 15:04 — page 98 — #8 ✐ 98 ✐ CHAPTER 8. Napoleon’s Triangles Since 6.K T / D Ta C Tb C Tc 3T , the inequality K T is equivalent to Weitzenb¨ock’s inequality Ta C Tb C Tc 3T . The relationship in (8.2) is actually stronger than the Weitzenb¨ock inequality, and enables us to prove the Hadwiger-Finsler inequality: If a, b, and c are the sides of a triangle, then Ta C Tb C Tc 3T C Tja bj C Tjb cj C Tjc aj : For a proof, see [Alsina

into their design. But often ellipses were approximated by figures using four or more circular arcs. Sebastiano Serlio (1475–1554), in his classic work Tutte l’Opere d’Architettura, describes four such constructions, one of which is based on the vesica piscis. Serlio recommended it for its simplicity, beauty, and ease of construction. To the pair of interesting circles we add two circular arcs, using the vertices of the vesica as centers and the diameter of the circles as radii, as shown in

of the shaded square and rectangle in Figure 1.2b are equal. He similarly reaches the same conclusion for the triangles in Figure 1.2c and the shaded square and rectangle in Figure 1.2d, which establishes the theorem. ✐ ✐ ✐ ✐ ✐ ✐ “MABK018-01” — 2011/6/1 — 17:14 — page 3 — #3 ✐ 1.1. The Pythagorean theorem—Euclid’s proof and more (a) (b) (c) ✐ 3 (d) Figure 1.2. In Figure 1.3 we see a modern dynamic version [Eves, 1980] of Euclid’s proof, where area-preserving transformations change