Pythagorean Theorem (Another trigonometric proof from New Orleans)
Also in other media:
- Article in NCTM - National Council of Teachers of Mathematics journal Mathematics Teacher: Learning & Teaching PK–12 (August 2024 edition)
- YouTube video walkthrough
In the GeoGebra construction below:
The left/right arrow buttons walk through each step of the proof. Click/tap once to start an animation to the adjacent step. Click/tap a second time to skip the animation and jump immediately to the adjacent step.
Alternatively, the point on the accompanying number line may be dragged.
This proof of the Pythagorean Theorem is more algebra-intense than many other proofs, so color coding is turned on my default to help show how equations connect to the diagram and how some equivalent expressions get substituted into others. The button in the upper-left corner toggles the color-coding on/off.
This GeoGebra construction presents my trigonometry-based proof of the Pythagorean Theorem. It was inspired by Calcea Johnson & Ne'Kiya Jackson, two remarkable high school seniors (St. Mary's Academy class of 2023) from New Orleans East. Their ingenious trigonometry-based proof of the Pythagorean Theorem hit the news cycle in March 2023. I don’t know them, but as a New Orleans school teacher, I admire and applaud their achievements. Here's my GeoGebra construction of what is believed to be their proof.*
Trig-based proofs of the Pythagorean Theorem are very rare, and I wanted to see if I could develop one too. I dabbled quite a bit during several days in April 2023 when my students were busy taking standardized tests. My breakthrough finally occurred while I was on hall-monitor duty on the last day of testing. The other monitor of that hallway was an art teacher who I didn’t assume would find my feat interesting, so I savored all the excitement of the moment by myself.
I had/have no desire to try to upstage Calcea Johnson & Ne'Kiya Jackson. Fittingly, I believe any objective observer would agree that their proof is more elegant than mine, but I’m proud to share my proof nonetheless. I duplicated the first couple steps from Johnson & Jackson’s work, but after that I veered off with an approach that I believe is original. The algebra-heavy nature of my subsequent steps perhaps offers greater assurance that I didn’t just follow an already-traveled path (This is in contrast with some earlier efforts, for which shorter and prettier proofs that I “discovered” were then found to be already documented elsewhere on the internet.)
Of course, with hundreds of published proofs of the Pythagorean Theorem out there in the world, one must be wary of guaranteeing completely unique work.
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* October 2024 update: Johnson and Jackson's proofs are published here.
![[color=#999999]sketch from the paper where this proof first emerged[/color]](https://www.geogebra.org/resource/cyaha3kq/Ec2EYmsWRHeCr9hY/material-cyaha3kq.png)