# Two High Schoolers Found an ‘Impossible’ Proof for a 2,000-Year-Old Math Rule—Then, They Discovered Nine More

Ne’Kiya Jackson and Calcea Johnson of Louisiana published a new study proving the Pythagorean theorem using trigonometry, a feat mathematicians long thought could not be done

In December 2022, a high school in Louisiana challenged its students with a bonus math question that came with a $500 cash prize: Using trigonometry, they had to create a new proof of the Pythagorean theorem, the 2,000-year-old rule that explains the relationship between the lengths of sides on a right triangle. No one told the teenagers that mathematicians had long considered this task “impossible.”

So, when Ne’Kiya Jackson and Calcea Johnson came up with a solution, it sent shockwaves through the math world. They presented their proof a few months later at the American Mathematical Society in 2023, but it still needed further review. This week, the pair of students, who are now in college, have published their findings in a new study—along with additional proofs of the theorem.

“Perhaps no subject in mathematics generates more confusion and anxiety for high school students than trigonometry,” Jackson and Johnson write in their paper. “There were many times when both of us wanted to abandon this project, but we decided to persevere to finish what we started.”

The research published on Sunday in *The American Mathematical Monthly** *outlines “five or ten” new ways to prove the Pythagorean theorem using trigonometry. That is to say, each of their proofs demonstrates that the square of a right triangle’s longest side equals the sum of the squares of the two shorter sides.

Jackson and Johnson spent months coming up with their original proof, even working on weekends and holidays. In the new paper, they describe that achievement and four new proofs that they worked out afterward. They also outline a new method that could result in five additional proofs.

“Some people have the impression that you have to be in academia for years and years before you can actually produce some new mathematics,” says Álvaro Lozano-Robledo, a mathematician at the University of Connecticut, to *Science News*’ Nikk Ogasa. But Jackson and Johnson’s work demonstrates that “you can make a splash even as a high school student,” he adds.

Although the Pythagorean theorem has been proved with algebra and geometry, mathematicians previously thought that it couldn’t be proved using trigonometry. That’s because the fundamentals in trigonometry are based on the same theorem, so any proof of it, experts suggested, would have to use principles that already assume the theorem is true, resulting in a logical fallacy called “circular reasoning.”

Instead, the pair of students used a trigonometric rule called the Law of Sines, which does not rely on the Pythagorean theorem. Two people had previously proved the theorem using trigonometry: Jason Zimba in 2009 and Nuno Luzia in 2015.

“I didn’t think it would go this far,” says Jackson in a statement. “I was pretty surprised to be published.”

Jackson and Johnson first worked on their proofs at St. Mary’s Academy in New Orleans, one of the oldest Catholic schools for Black women in the country. After presenting their work, the pair set out to publish their findings in a peer-reviewed journal, and they worked on the past proofs of the theorem. This “sparked a creative process,” Jackson tells *Science News*, adding that from there they developed the additional proofs.

Now, Jackson is pursuing a doctoral degree in pharmacy at Xavier University of Louisiana and Johnson is studying environmental engineering at Louisiana State University.

The duo’s achievement showed how a mere bonus question at a high school can lead to surprising mathematical discoveries, even ones that were previously thought to be “impossible.”

“Mathematicians, I think, have learned to not make a bold claim that something is impossible, because we’ve been embarrassed over the years too many times by doing that,” Stuart Anderson, a mathematician at Texas A&M University–Commerce, told *Scientific American*’s Leila Sloman last year, after Jackson and Johnson first presented their work.

Anderson, who is a retired professor, hopes this new proof raises interest in mathematics among students. “It kind of makes me wish I still had a class so I could talk about it,” he tells *Scientific American*.

This enthusiasm is echoed by Johnson, who highlights the importance of representation in the field. In the statement, she says, “I am very proud that we are both able to be such a positive influence in showing that young women and women of color can do these things.”