I don't think it's quite circular, but in a sense you're right.

if c = -a + b, then for any inner product you have

<c,c> = <a,a> + <b,b> - 2<a,b> as a result of bi-linearity of the inner product.

By definition if <a,b> = 0 then a and b are orthogonal and a version of pythagoras' theorem holds.

But you're right in saying the choice of inner product

<x,y> = x_1y_1 + ... + x_ny_n

is motivated by pythagoras' theorem in Euclidean space.

By definition?

In the abstract world of algebra we are free to choose any definition (different rules will give different algebras).

But if we want our algebraic manipulations to prove the theorem about geometric objects we need to prove isomorphism between our algebra and the geometric objects and operations on them.

I doubt distributivity and other properties of operations on geometric vectors can be proven without the the Pythagorean theorem.

> By definition?

Yes, in a Hilbert space (i.e. an abstract vector space with an inner product), the definition of orthogonality is that the inner product of two nonzero vectors is zero.

I'm not sure I really know what you mean by geometric vectors.

Well... think harder, what can I say :)

From that comment I'm not sure you do either.

I suspected trolling in your question about geometric vectors.

Sides of a triangle and elements of your algebra are different domains. In order to translate results between them one needs to prove this makes sense.

In the article the author only shows that inner product of c by itself equals to sum of inner product squires of a and b, if a and b are orthogonal.

Who told you this has anything to do with lengths of triangle sides?

there are hundreds of proofs for the pythagorean theorem, surely some of them using the axioms of geometry

This isn't just using the axioms of geometry, though, it's trying to prove the Pythagorean theorem using the Pythagorean theorem as an axiom. Hence circularity.

no, not exactly. the euclidean norm can be defined to fulfill a couple basic properties or equations (like composibility, or schwarz' inequality, but I don't recall exactly), and it's pure coincidence, if you will, that the norm is equal to the root mean square. That's not circular reasoning.

The name euclidean norm implies that the geometric angle (no pun intended) was the motivation, but what's really central is a question of epistemology. There's not much of a point to prescribe a certain approach over another, without a good argument.