Two students who discovered a seemingly impossible proof to the Pythagorean theorem in 2022 have wowed the math community again with nine completely new solutions to the problem.
While still in high school, Ne'Kiya Jackson and Calcea Johnson from Louisiana used trigonometry to prove the 2,000-year-old Pythagorean theorem, which states that the sum of the squares of a right triangle's two shorter sides are equal to the square of the triangle's longest side (the hypotenuse). Mathematicians had long thought that using trigonometry to prove the theorem was unworkable, given that the fundamental formulas for trigonometry are based on the assumption that the theorem is true.
Jackson and Johnson came up with their "impossible" proof in answer to a bonus question in a school math contest. They presented their work at an American Mathematical Society meeting in 2023, but the proof hadn't been thoroughly scrutinized at that point. Now, a new paper published Monday (Oct. 28) in the journal American Mathematical Monthlyshows their solution held up to peer review. Not only that, but the two students also outlined nine more proofs to the Pythagorean theorem using trigonometry.
Pythagorean theorem has over 100 proofs, they are just geometric one, this is the first trigonometric one. The reason it is impressive is that the pythagorean theorem is foundational to trigonometry, so any attempt at a trigonometric often calls on the pythagorean theorem implicitly.
One might need to prove that a trigonometric proof isn't equivalent to any geometric proof. Somehow the premise here "a proof based on the sine law" doesn't inspire that confidence for me as sine law has equivalent formulations in geometry.
That said, I'd also say that the boundary between geometry and trigonometry isn't a particularly necessary one, and the work of these young girls exposes this.