For thousands of years, mathematicians thought it was nearly impossible to prove the Pythagorean Theorem using trigonometry without assuming that the theorem itself was true. But as one pop star once taught us: Never say never.

**Two high schoolers take on a 2,000-year-old theorem**. Back in 2022, Louisianna high school students Calcea Johnson and Ne’Kiya Jackson discovered a new proof for the Pythagorean Theorem after they entered a local math competition. Put simply, a math proof is an explanation for why a math statement is always true.

“The bonus question was to create a new proof of the Pythagorean theorem. Motivated by the $500 prize, we independently decided to take on this task,” Johnson and Jackson write in the paper, noting that it was a lot harder than they thought. “After roughly a month of mental labor, we each completed and submitted our work.”

The Pythagorean Theorem, named after the Greek philosopher and mathematician Pythagoras, is a staple in math classes and says the following: a² + b²= c². In other words, the longest side of a triangle, known as the hypotenuse, is equal to the sum of the squares of the two shorter sides. As such, it means that you can determine the length of any side of right triangle if you know the length of two of the other three sides.

**Nearly impossible. **While many mathematicians have proven the Pythagorean Theorem over the years using algebra and geometry, few have done so using trigonometry, a branch of math focused on the relationships between the sides and angles of triangles. According to experts, this is because the field of trigonometry is essentially based on the Pythagorean Theorem.

Consequently, mathematicians solving the Pythagorean Theorem using trigonometry often resort to circular reasoning, which essentially means assuming the theorem is already true. This is a logical failure because it doesn’t explain why the theorem is true.

However, Johnson and Jackson managed to prove the Pythagorean Theorem without using circular reasoning in 2022, something that has only been done twice before by professional mathematicians.

**10 new proofs**. Johnson and Jackson published their work in the newest issue of the journal *American Mathematical Monthly** *this week. The paper, which has been peer-reviewed, includes their original proof as well as 9 additional proofs. The pair’s discovery led to a wave of recognition, including a congratulatory message from Michelle Obama. None of their proofs rely on circular reasoning.

Johnson and Jackson expressed being both excited and surprised that they had gotten published for their work, especially at their age.

“It’s very exciting for me, because I know when I was growing up, STEM wasn’t really a cool thing,” Johnson said in a statement. “So the fact that all these people actually are interested in STEM and mathematics really warms my heart and makes me really excited for how far STEM has come.”

Both women are now in college. Johnson is studying environmental engineering at Louisiana State University. Jackson is currently at the Xavier University of Louisiana and is working towards a doctorate in pharmacy.

**A new set of eyes**. The work of the two young women is a good reminder to question what others have considered impossible. Tom Murdoch, an honorary math professor at the University of Bristol in the UK, called the study “intriguing” in an interview with *CNN*.

“I think sometimes having very little knowledge of the problem means that you are not bound by what’s gone before. Looking at this with fresh eyes, which I think they’ve done, that’s the really impressive thing,” Murdoch said.

Image | Saad Ahmad

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