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Of all the figures in plane geometry, the triangle is the most interesting, and the most prolific in terms of producing theorems. Moreover, of all the triangles, the equilateral seems to stand out as perfection personified.
J. Garfunkel and S. StahlEquilateral triangles lie at the heart of plane geometry. In fact Euclid's first proposition–Proposition 1 in Book I of the Elements–reads [Joyce, 1996]: To construct an equilateral triangle on a given finite straight line. Equilateral triangles continue to fascinate professional and amateur mathematicians. Many of the theorems about equilateral triangles are striking in their beauty and simplicity.
Mathematicians strive to find beautiful proofs for beautiful theorems. In this chapter we present a small selection of theorems about equilateral triangles and their proofs.
Pythagorean-like theorems
The Pythagorean theorem is usually illustrated with squares on the legs and hypotenuse of the triangle, and many lovely visual proofs employ such illustrations. See Section 5.1 for several examples. However, as a consequence of Proposition 31 in Book VI of the Elements of Euclid, any set of three similar figures can be used. See Figure 6.1, where in each case the sum of the areas of the shaded figures on the legs equals the area of the unshaded figure on the hypotenuse.
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