TITLE: Proof of the Pythagorean Theorem without using the concept of area?
QUESTION [7 upvotes]: Most of the proofs of Pythagorean Theorem that I see all seem to involve the concept of area, which to me does not seem "trivial" to prove. 
Others show proof for a particular triangle but it does not seem clear to me if it works for all right triangles or just specific variants. 
Is there a proof that is purely algebraic based on algebraic triangle constraints? Or one that does not rely on area at least and works for any arbitrary right triangle?

REPLY [4 votes]: I realize this question is old, but I wonder if the OP would have been OK with the following:
Let $\triangle ABC$ be a right triangle with $\angle ACB$ the right angle.  Drop an altitude from $C$ to $\overline{AB}$ at $D$.  Then $\triangle DCA$ and $\triangle DBC$ are both right triangles and similar to $\triangle ABC$.

By similarity,
$$
\frac{AD}{AC} = \frac{AC}{AB}
$$
and hence
$$
AD = \frac{AC^2}{AB}
$$
Similarly (!),
$$
\frac{DB}{BC} = \frac{BC}{AB}
$$
and therefore
$$
DB = \frac{BC^2}{AB}
$$
Finally,
$$
AB = AD+DB = \frac{AC^2}{AB}+\frac{BC^2}{AB}
$$
leading directly to
$$
AB^2 = AC^2 + BC^2
$$
as desired.
I'm not sure this avoids things that are fundamentally equivalent to assuming area, but perhaps it would have been satisfactory to the OP?