Algebraic Proof of the Pythagorean Theorem

The area of square ABCD can be calculated in two ways.

The first method is squaring its side length, $a + b$, giving $(a + b)^2$.

The second method is adding the areas of four congruent right triangles and the central square.
Each triangle has an area of $\frac{1}{2}ab$, so multiplying this by $4$ gives $2ab$ for the total area of the four triangles. Adding the central square’s area, $c^2$, gives $2ab + c^2$.

Equating the two expressions:
$(a + b)^2 = 2ab + c^2$

Simplifying yields:
$c^2 = a^2 + b^2$

which proves the Pythagorean theorem.