While it’s very impressive, and two high school teens should absolutely be celebrated for such an accomplishment, it’s more of a mathematical curiosity. It doesn’t have like a significant impact. I don’t say that to diminish their achievement, just noting it’s not like a revolution that challenges the world.

The gist is that, for 2000+ years it was assumed that a trigonometric proof of the theorem is going to boil down to using laws that are in some way *derived* from the Pythagorean theorem. You can’t use the premise in a proof like this. Through some law of sines shenanigans apparently they’ve discovered a succinct proof that *doesnt* violate this. Notable mathematicians have historically been cited as commenting offhand “it shouldn’t be possible”, but these two figured out a method that hadn’t occurred to anyone. I also read they’ve extended the technique to other “missing” classic proofs.

Neither is pursuing math in university! Hopefully they’ll go on to be titans in their fields, whatever they do.

We already knew that the Pythagorean theorem was true, in fact it’s been proved in a zillion different ways. However, it was believed for over a century that you could not derive a^2 + b^2 = c^2 from trigonometry, because it was thought that you’d need the law of cosines to do it…which is *built upon* the Pythagorean theorem. That would be a circular proof.

What Jackson and Johnson’s proof showed was that you *do not* need the law of cosines to do this. You can get away with just using the law of sines, which is completely independent of the Pythagorean theorem.

In terms of new knowledge gained, there wasn’t much. What this proof really did was show that mathematicians, as humans in a social group, had accepted some received wisdom from a respected past mathematician, rather than questioning it and finding the (fairly straightforward) proof that was allegedly so “impossible.” Developments like this, where a previously-unconsidered pathway is revealed, are prime candidates for revolutionary new mathematics. That wasn’t the case *this* time, but it could be for a future example.

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