When we compare two plane figures, we are not only interested in whether they are equal in the sense of “superimposable” (that is, congruent): often we want to know whether they occupy the same surface.
Definition — Equivalent figures
Two figures are said to be equivalent when they have the same area.
Two congruent figures are always equivalent, but the converse does not hold: a rectangle and a triangle can have the same area while having completely different shapes.
The most elegant way to prove that two figures are equivalent is equidecomposability: one cuts a figure into a finite number of pieces and reassembles them, without overlaps or gaps, until the other figure is obtained. Since the total area does not change under cutting and reassembling, two equidecomposable figures are necessarily equivalent. It is precisely this simple yet extremely powerful idea that we shall use to prove Pythagoras’ theorem.
Links
Topics: Equivalence and Pythagoras
Concepts: Area · Equidecomposability · Equivalence
Methods: Equivalence of figures
Skills: Synthetic geometry