It's fully symmetric. Basic arithmetic covers the following identities:
1. (a² - b²) = -(b² - a²), because of even powers
2. (a - b) = -(b - a)
So the following two statements are the same statement:
3. (a² - b²) = (a - b)(a + b)
4. -(b² - a²) = -(b - a)(b + a)
Let's assume this only holds for a>b (because we're content that the geometric proof shows that):
3a. (a² - b²) = (a - b)(a + b), a > b
But we don't know if it also holds for b>a... after all, how would you show a negative areas? What does that even mean Turns out: it doesn't matter, the b>a relation reduces to the same formulae as the a>b relation, so the geometric proof covers both. To see why, some more elementary algebra: we can invert both sides of (4), provided we also invert the relation between a and b, so this:
4a. -(b² - a²) = -(b - a)(b + a), b > a
Is the same as this:
4b. (b² - a²) = (b - a)(b + a), a > b, by inversion
Of course, algebra doesn't care about which labels you use, as long as the identities and relations between them are preserved, so we can swap "a" for "b" and "b" for "a" in both the identity and relation in (4b) to get:
4c. (a² - b²) = (a - b)(a + b), b > a
And we found a symmetry that we (maybe) didn't realize was there:
3a. (a² - b²) = (a - b)(a + b), a > b
4c. (a² - b²) = (a - b)(a + b), b > a
Same formula, inverse relation. It turns out that it doesn't matter whether we start with a>b or b>a, they reduce to the same expression, thanks to those even powers, and a geometric proof for one is by definition a proof for the other.
That's the point. The geometric proof requires that you show the applicability for the b>a case with algebra. If you don't, it's not complete. And if you do, you can just show everything with algebra in the first place, and shorter, and also for a,b element C (instead of R), at the same time.
I'd have to disagree - it is perfectly fine to show a visual proof and simply state that we can reduce b>a to a>b and this proof therefore covers both, optionally with a little "I don't believe you, show me the math" pop out.
The visual proof is the neat part that people literally can't think of unless you show it to them, after which things might suddenly click for them. The algebraic proof is boring AF and doesn't make for a good maths hook ;)
