Two figures are said to be equivalent when they have the same area. The most elegant way to prove equivalence is equidecomposability: cutting a figure into pieces and reassembling them to obtain the other. With this simple yet extremely powerful idea, we shall prove the most famous theorem in elementary geometry: Pythagoras’ theorem, according to which the square built on the hypotenuse of a right-angled triangle is equivalent to the sum of the squares built on the legs. The chapter closes with a historical look at the Pythagorean school and at the discovery of the irrationals, one of the decisive moments of mathematical thought.
Sections
- Equivalence and equidecomposability
- Equidecomposability of a triangle and a parallelogram
- Pythagoras’ theorem