2 La. H.S. students complete an ‘impossible’ proof

Mathematicians had long thought that proving the Pythagorean Theorem using trigonometry was impossible. Then, a Louisiana high school held a competition to award $500 to a student or students who could do it. Two did, the Smithsonian Magazine reports.

Ne’Kiya Jackson and Calcea Johnson were awarded keys to the city of New Orleans and even received personal praise from Michelle Obama after astonishing their teachers in December 2022. Their proof has since been reviewed by peers and published in The American Mathematical Monthly.

The two have since moved on to college: Ms Johnson is studying environmental engineering at Louisiana State University, and her high school classmate, Ms Jackson, is now a student in pharmacology at Xavier University of Louisiana.

The theorem, which allows one to determine the length of a side of a right triangle given the lengths of the other two sides, has been around for some 2,000 years. It has been proved using algebra and geometry, but a proof using trigonometry had eluded mathematicians for centuries.

Their article is entitled “Five or Ten New Proofs of the Pythagorean Theorem” because, after they discovered the first trigonometric approach to the proof, they were led to four more and then, quite possibly, another five.

Diagram of their first proof

Mathematicians previously thought the theorem couldn’t be proved using trigonometry because the foundations of trigonometry depend on it. Therefore, any proof that used trigonometry would have to rely on statements that already assume the theorem is true, a logical fallacy called “circular reasoning.”

But instead of using statements that assumed the theorem was already true, the two students at St Mary’s Academy in New Orleans, one of the oldest Catholic schools for Black women in the country, used a trigonometric rule called the Law of Sines, which doesn’t rely on the Pythagorean theorem.

Although the proofs Ms Jackson and Ms Johnson discovered are new, two people had previously proved the theorem using trigonometry: Jason Zimba in 2009 and Nuno Luzia in 2015.