ARTICLES

# The Pythagorean theorem

In this post I will provide an intuitive proof of a very foundational little result in elementary geometry: the Pythagorean theorem.

**Statement**
Given a right triangle as the one in *Figure 1.i* with legs *a* and *b* and hypothenuse *c*, the Pythagorean theorem states that the area of the square
whose side is *c* is the sum of the areas of the two squares having sides *a* and *b*. See *Figure 1.ii*.

*Figure 1*

In formulas the theorem is saying that:

$$ c^2 = a^2 + b^2 $$

**Proof**
*Figure 2* will help us constructing a proof.

*Figure 2*

The triangle in *Figure 2.i* has area *ab / 2*. This is because it is exactly half of a rectangle with base *a* and height *b* (see *Figure 2.ii*).
Arranging four copies of the triangle in the way shown in *Figure 2.iii* defines two squares; a smaller square with side *c* and a large square
with side *a + b*. The area of the big square with side *a + b* is equivalent to the area of the small square with side *c* plus the area of the
four copies of the triangle. In formulas:

$$ c^2 + 4 \frac{a b}{2} = (a + b)^2 $$

Let’s expand the right term of the equation and simplify the left term:

$$ c^2 + 2 a b = a^2 + 2 a b + b^2 $$

By subtracting *2ab* from both terms we obtain:

$$ c^2 = a^2 + b^2 $$

Thereby proving the statement.