[layer8]

The geometric “proof” isn’t actually equivalent, because it assumes a > b, and doesn’t generalize geometrically to b > a. The algebraic proof, on the other hand, generalizes at least to commutative rings.

Geometric “proofs” like this are neat, but are no real substitute for the algebraic ones. I’d argue that in cases like the present one they also don’t provide any deeper insights. You’re just moving geometric shapes around instead of algebraic symbols. They might give you the feeling that the theorem isn’t as arbitrary as you thought, but it isn’t arbitrary in algebra either.

I’m putting “proof” in quotes here because there are many examples of incorrect geometric “proofs”, and there is generally no formal geometric way to verify their correctness.

[zozbot234]

> there is generally no formal geometric way to verify their correctness.

There are formal models of synthetic (i.e. axiom-and-proof based) Euclidean geometry where proofs can in fact be verified. This is accomplished by rigorously defining the set of allowed "moves" in the proof and their semantics, much like one would define allowed steps in an algebraic computation.

[layer8]

Yes, but once you work with those formal models, you’re really doing algebra and not intuitive geometrics anymore.