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.