A proof of the Pythagorean theorem using "rearrangement"


The Pythagorean theorem, or Pythagoras' theorem is a relation among the squares constructed on the sides of a right triangle (or right-angled triangle).

In his monumental book "The thirteen books of Euclid's Elements, volume I: Books I-II" (Dover Publications, second edition, 1956), augmented with many mathematical and historical annotations, Thomas L. Heath states proposition I47, better known as the Pythagorean Theorem as follows:

Proposition I47

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.



As the approach is purely geometrical, no algebraic expressions are used. "Equality" between figures (line segments, triangles, parallelograms, rectangles, triangles... ) is first expressed in the sense of congruence. In some propositions preceeding I47 the notion of equality is extended to some figures that do not need to be congruent!

Proposition I36

Parallelograms which are on equal bases and in the same parallels are equal to one another.




Proposition I38

Triangles which are on equal bases and in the same parallels are equal to one another.





Besides congruence and equality of figures in accordance to I36 and I38, the proof of the Pythagorean theorem I47 in the Elements also uses some of the so-called "Common notions":

1. Things which are equal to the same thing are also equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.

Hundreds of proofs of the Pyhagorean theorem exist. Some of these proofs use algebraic formulas. Other proofs, such as the original proof of I47 in the Elements, are purely geometrical. They compare areas without using algebraic formulas.

One of the most simple proofs only uses a rearrangement of triangles within a square and the "Common notions". In the following image, the large square on the left consists of the square on the hypothenuse and four times the right triangle, the same large square on the right is composed of the squares on the two other sides of the right triangle and also four times the tight triangles. Using common notion 3, we see that the the red square on the left equals the two red squares on the right. And that's what Pythagoras' theorem states!
The animation demonstrates the rearrangement of the triangles.



This approach is purely geometrical. If we allow the use of algebraic expressions for the length of a line segment and the area of a square, it's clear that the Pythagorean theorem for a right triangle with a and b as the lengths of the rectangular sides and c as the length of the hypothenosa is expressed in the well-know algebraic formula:

a2 + b2 = c2




On YouTube I posted another animation about the Pythagorean Theorem, accompanied by a song... CLICK HERE
Herman Serras (update December 2025)
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