There's some media reporting that two high school students, Calcea Johnson and NeKiya Jackson, have proven the Pythagorean Theorem using the Law of Sines, the first proof using trig without using circular reasoning. (https://www.insider.com/us-teens-claim-to-have-proved-pythagorean-theorem-thought-impossible-2023-3?amp) Apparently such a feat was thought impossible; The Pythagorean Proposition by Elisha Loomis says that “There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem.”
They've presented their proof at the American Mathematical Society, who suggest they submit it to a journal. The proof hasn't been verified by experts, nor did I have success finding it online to check myself.
Resolves to the consensus among the expert mathematics community in three months (June 29, 6pm PT) about whether this proof is essentially valid, trig-based, and free from circular logic. Can still resolve YES if it has an easily fixable flaw. If there's no clear consensus in the math community, I'll resolve to 25%, 50%, or 75% based on the approximate split.
General policy for my markets: In the rare event of a conflict between my resolution criteria and the agreed-upon common-sense spirit of the market, I may resolve it according to the market's spirit or N/A, probably after discussion.
Mar 29, 8:01am: Will experts find the non-circular trig “proof” of the Pythagorean Theorem by high schoolers legit? → Will experts find the trig-based “proof” of the Pythagorean Theorem by high schoolers legit?
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This is clearly legit, see https://www.youtube.com/watch?v=p6j2nZKwf20 which I saw on my own. Upon seeing this on manifold, i was surprised to see it so low, why are people still buying "No"?
@JosephNoonan I'm still skeptical that the "calculus" part of the proof doesn't contain the Pythagorean theorem somehow. I believe that we can construct the "waffle cone triangle" from the upper left side and the two angles. I then believe we can construct the inductive sequence of points on either side which define the new triangles, by repeatedly taking horizontal or vertical lines from the point we have and intersecting them with the other line. How do we prove, though, that this sequence of points converges to the lower right tip of the waffle cone triangle?
@BoltonBailey I guess we can do another law of sines invocation to show that for the triangles made by the lower right tip and two ends of a horizontal or vertical line have a ratio less than 1 between their two long sides, so the sides of these triangles are geometrically decreasing to zero?
@BoltonBailey I guess this is ok, but there's also the awkward matter of when alpha is 45 degrees, and then this waffle cone goes on forever, and the argument breaks down. So to be pedantic, this only proves the Pythagorean theorem on the precondition that the triangle is not a 45-45-90 triangle.
@BoltonBailey Yeah, I don't think this corner case should undermine the proof being trigonometric as applied to almost all triangles?
I guess one could also check what their official fix is for alpha being 45 degrees; feels like it should be something simple.
@BoltonBailey I can't necessarily think of a simple fix that isn't just using a standard non-trig proof for the special case.
@BoltonBailey Thanks for trying! I think I will just ignore this issue for resolution though, seeing as 0% of right triangles are isosceles.
I think I will just ignore this issue for resolution though, seeing as 0% of right triangles are isosceles.
Not a good reason to ignore it, IMO. I can prove to you that all numbers in the interval [0,1] are rational, except for 0% of them, but that's not a good reason to ignore that 0%. However, in this case, I think the issue should be ignored, since there is only one case not covered by the proof, and that case is trivial to verify on your own. Also, the high-schoolers' write-up might have treated the case anyway, we don't know whether they mentioned it or not from the video.
@Conflux Plus, this case can still be treated along the same lines as the high-schoolers' proof, without even using the waffle cone. Since a = b and α=π/4 in this case, a^2+b^2 = 2ab/sin(2α) reduces to sin(π/2) = 1, which is of course, true, and doesn't require the Pythagorean theorem to prove. That's the identity that they needed the waffle cone for in their proof, and in the case of α=π/4, they don't even need calculus, so it really is purely trigonometric.
@Conflux You said you would resolve it per
> Resolves to the consensus among the expert mathematics community in three months
and the consensus is that the proof is legit. If the people here think they know better about the supposed "edge case" then they should file an issue with the authors.
@JosephNoonan Yeah, good point about the 0%, and thanks for the way to extend their proof.
@rigille Yep, seems to confirm things. If this remains the mathematician consensus, this will resolve YES. Pretty cool for the high schoolers! Even though it’s not the first trig-based proof.
Just found this vid which reconstructed the proof from images of the slides and explained it. Seems real, right? https://m.youtube.com/watch?v=nQD6lDwFmCc&feature=emb_logo But idk, maybe there’s a subtle flaw
@Conflux The proof also appears to be noticeably distinct from the trig-based proofs listed at https://www.cut-the-knot.org/pythagoras/TrigProofs.shtml as well as the series proof at https://www.cut-the-knot.org/pythagoras/Proof100.shtml (hat tip to @drevv for links to those resources). The proof is arguably a bit more complex than either of those, but still big kudos to authors!
Since this question asks only about their "proof", I think this is very undervalued currently. They appear to rely on convergent series in their proof (like in this proof: https://www.cut-the-knot.org/pythagoras/Proof100.shtml) and Zimba showed trig proofs are possible: https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf. It was already clear the Loomis statement was incorrect (via Zimba), but the students being wrong about this (i.e. being unfamiliar with the Zimba work) does not imply their proof is incorrect.
@NicoDelon I’m assuming that if it’s trivial, it will involve circular reasoning. But “legit” entails, like, it’s a breakthrough like they’re claiming
@NicoDelon wait, I just realized this contradicts what I said in the description, which is just that the proof had to be “valid, trig-based, and free from circular logic”, not necessarily a breakthrough. I’ll send you a manalink if you want
