Proving the Pythagorean theorem

We will now prove the Pythagorean theorem:

Statement: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Specifically, if a and b are the legs and c is the hypotenuse, then a 2 + b 2 = c 2 a^2 + b^2 = c^2 .

Proof: We can prove the theorem algebraically by showing that on this figure the area of the big square equals the area of the inner square (hypotenuse squared) plus the area of the four triangles:

( a + b ) 2 = c 2 + 4 ⋅ ( 1 2 a b ) a 2 + 2 a b + b 2 = c 2 + 2 a b a 2 + b 2 = c 2 \begin{align*} (a + b)^2 &= c^2 + 4 \cdot \left( \frac{1}{2} ab \right) \\ a^2 + 2ab + b^2 &= c^2 + 2ab \\ a^2 + b^2 &= c^2 \end{align*}