Using Similarity to Prove the Pythagorean Theorem

The Pythagorean Theorem is one of the most interesting theorems for two reasons: First, it’s very elementary; even high school students know it by heart. Second, it has hundreds of proofs. The proof below uses triangle similarity.

Pythagorean Theorem

In a right triangle with side lengths $a$ and $b$ and hypotenuse $c$,  the following equation always holds:

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

Proof

In the figure above, triangle $BDA$ is similar to triangle $ADE$.  In similar triangles, the corresponding sides are proportional. It follows that

$\displaystyle\frac{BD}{AD} = \frac{AD}{DE}$

which is equivalent to

$\displaystyle\frac{c + a}{b} = \frac{b}{c-a}$.

Cross multiplying, we have $c^2 - a^2 = b^2$. Adding $a^2$ to both sides of the equation, we  have $c^2 = a^2 + b^2$, which is what we want to prove.

2 thoughts on “Using Similarity to Prove the Pythagorean Theorem”

1. Nice proof! I’ve seen a version of this one where the hypotenuse of the right triangle is the diameter of the circle. I don’t think I’ve seen the idea of using the hypotenuse as the radius like this.

• It’s actually a derivative of one of the proofs I’ve seen. I’m searching for the same proof on the net. It could be an addition to the hundreds of proofs.