Calcea Johnson and Ne’Kiya Jackson are two high school students at St. Mary's Academy in New Orleans who recently presented a new proof of the Pythagorean theorem at the Spring Southeastern Sectional Meeting of the American Mathematical Society. There are, of course, many proofs of the Pythagorean theorem, but what sets this one apart is that it's (mostly) trigonometric, but does not circularly rely on the Pythagorean identity. You can read the AMS abstract here (https://meetings.ams.org/math/spring2..., and you can read more about these remarkable young women here (https://www.theguardian.com/us-news/2....

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There's not a lot of information online yet about how their proof proceeds, but from a few of the slides you can pull out of news coverage (you can see de-skewed versions of three of them here: https://www.dropbox.com/s/h99ezl8l360..., it seems the proof relies roughly on the definition of the sine ratio itself and the Law of Sines, along with a very clever "waffle cone" triangular shape.

If you'd like to play around with the waffle cone shape, I built a little illustration in Desmos that you van view here: https://www.desmos.com/calculator/tyw.... You can see the interplay of the infinitely many right triangles going down to the right from the isosceles triangle, and it also includes a folder with the proof.

Finally, some news coverage has gone a little overboard on stating the accomplishment here. It is true that a collection of proofs of the Pythagorean theorem from roughly a hundred years ago stated that a trigonometric proof was impossible (Elisha Loomis wrote "no trigonometric proof is possible" here: https://files.eric.ed.gov/fulltext/ED.... But in 2009, Jason Zimba published just such a result in Forum Geometricorum: https://forumgeom.fau.edu/FG2009volum.... You can see that proof and other related proofs at Cut the Knot:

+ Jason Zimba's proof: https://www.cut-the-knot.org/pythagor...
+ More Trigonometric Proofs: https://www.cut-the-knot.org/pythagor...
+ Nuno Luzia's Half-Angle Proof: https://www.cut-the-knot.org/pythagor...
+ John Arioni's Convergent Geometric Series Proof: https://www.cut-the-knot.org/pythagor...

#pythagoreantheorem #pythagoras #impossibleproof #highschool

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