Pythagorean Theorem Proof, new in 2023
There has been a lot of local, national, and even international press lately for the remarkable New Orleans (Louisiana, United States) teens Calcea Johnson and Ne’Kiya Jackson. They have apparently discovered a new proof of the Pythagorean Theorem that is based on trigonometry without commiting any circular reasoning (i.e. without directly or indirectly using the Pythagorean Identity ). Kudos to them and anyone who has supported them in demonstrating how driven and accomplished they are.
Their work is not yet published,* so I hope I and other nerds who are curious for Mathy details can discuss and explore and speculate without detracting from the credit which Johnson and Jackson deserve.
The GeoGebra construction below is based on the speculation of this video by MathTrain, who I think did a good job of deducing how he believes Johnson's and Jackson's proof unfolds while giving them due credit throughout. Here is the local (here in New Orleans) WWL news story that included a glimpse of one of Johnson's and Jackson's presentation slides, upon which all this speculation is based. The first two photos below are screen-captures from that video.
As viewers of the MathTrain video were invited to do, I'd urge those with a Mathematically-adventurous spirit to look at the still images first and attempt to deduce the proof from there before interacting with the GeoGebra construction.
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* October 2024 update: Johnson and Jackson's proofs are published here.
Whenever you're ready to see the proof which is speculated to be that of Calcea Johnson and Ne’Kiya Jackson, use the slider below to see the steps.
Additional proof
Inspired by Calcea Johnson and Ne’Kiya Jackson, I set out to find another trigonometric proof of the Pythagorean Theorem. Here is the GeoGebra construction of the resulting proof I developed. I later found out from Johnson and Jackson (via their high school Math teacher) that they had already independently found a practically identical version of this proof along with several others!*
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* October 2024 update: In the publication of Johnson and Jackson's proofs, I don't see any work that looks particularly like my proof (other than the starting diagram of a reflected right triangle and use of law of sines, which I copied from their first publicized proof). I'm not sure where the miscommunication lies. I certainly don't mind being able to claim my proof as mine alone, and my admiration for Calcea Johnson and Ne’Kiya Jackson is only heightened in examining their work.