Pythagorean theorem does require the Cauchy-Schwarz inequality? In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved by the Cauchy-Schwarz inequality? Or effectively is the Pythagorean Theorem a stronger if not equivalent notion?
 A: $\newcommand{\ip}[2]{\left\langle #1,#2\right\rangle}$
$\newcommand{\norm}[1]{\left|\left| #1\right|\right|}$
I think your confusion is due to notation used in Pythagorean theorem: we don't actually require any norm-property of $||\cdot||$ induced by the inner product in proving it. It's only after we recognise $||\cdot||$ as a norm that it has its usual realization.
Consider an inner product space $V$ over real field for simplicity.
Let's denote 
$$
\norm{x}^2=\ip{x}{x}
$$
The important thing to realize here is that although notation $||\cdot||$ is used, at this stage we don't recognize $||\cdot||$ as a "norm". It's just a convenient notation to denote $\langle x,x\rangle$.
Suppose now that $x,y$ are orthogonal. We prove the "Pythagorean Theorem"

Claim : If $x,y$ are orthogonal then
  $$
||x+y||^2=||x||^2+||y||^2
$$
  Proof : $$\ip{x+y}{x+y}= \norm{x}^2+\norm{y}^2+2\ip{x}{y}=\norm{x}^2+\norm{y}^2$$ by orthogonality, bilinearlity.

So that didn't require CS inequality.
So this implies CS inequality, and this CS inequality ensures that the function $\norm{\cdot}$ is indeed a norm. In short,
\begin{align*}
IP &\implies \text{Pythagorean Theorem} \implies \text{Cauchy-Schwarz} \\
&\implies \text{"norm function" is indeed a norm}
\end{align*}
