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 [math]sin^2\theta+cos^2\theta=1[/math]). Kudos to them and anyone who has supported them in demonstrating how driven and accomplished they are.[br][br]Their work is not yet published,[color=#ff00ff]*[/color] 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.[br][br]The GeoGebra construction below is based on the speculation of [url=https://www.youtube.com/watch?v=nQD6lDwFmCc]this video by MathTrain[/url], 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) [url=https://www.youtube.com/watch?v=Ka1k4i1ueNU]WWL news story[/url] 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.[br][br]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.[br]________________________[br][br][color=#ff00ff]*[/color] October 2024 update: [url=https://www.tandfonline.com/doi/full/10.1080/00029890.2024.2370240]Johnson and Jackson's proofs are published here[/url].[br]

[size=150]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.[/size]

[size=150][b]Additional proof[/b][/size][br]Inspired by Calcea Johnson and Ne’Kiya Jackson, I set out to find another trigonometric proof of the Pythagorean Theorem. Here is the [url=https://www.geogebra.org/m/za5f6czt]GeoGebra construction of the resulting proof I developed[/url]. 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![color=#ff00ff]**[br][/color]________________________[br][br][color=#ff00ff]**[/color] October 2024 update: In the publication of [url=https://www.tandfonline.com/doi/full/10.1080/00029890.2024.2370240]Johnson and Jackson's proofs[/url], 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.

[b]More Pythagorean stuff[/b][br]Check out my GeoGebra implementations of some classic [url=https://www.geogebra.org/m/pavngm8p]Pythagorean Theorem Proofs Without Words[/url].