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Comment by orra | Hacker News Reader

> It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. [...] whereas the Greeks definitely did (we don't know whether Pythagoras himself did it

Wait, really? I thought the proofs back then were geometric, so only proved specific instances. And so it wasn't until algebra, maybe trig, was discovered it was properly proven?

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There are fully general and valid geometric proofs, eg http://mathandmultimedia.com/wp-content/uploads/2010/02/pyth...

Despite labels $a$, $b$, $c$, I don't see how it's fully general. At best you could say it works for similar triangles, if you have that concept.

But I don't see how it proves anything if $a$ were say doubled, and $b$ kept the same.

I'm not sure what you mean, sorry. It's definitely fully general. It demonstrates that:

* For any real positive values $a$ and $b$

* let $T$ be the (unique) right-angled triangle with non-hypotenuse sides of length $a$ and $b$

* then $a$^2 = $b$^2 = $c$^2

* where $c$ is the (unique) length of the hypotenuse of $T$.

You could give me the instance

  $a$ = 2, $b$ = 3

and it will prove it for this case. Then you could later double $a$ and give me the instance

  $a$ = 4, $b$ = 3

and it will prove it just as easily for this case.

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