What started as a bonus question in a high school math competition has resulted in no fewer than ten new ways to prove the age-old mathematical rule of the Pythagorean theorem.

It has long been argued that it is impossible to use trigonometry to prove what is in fact a theorem fundamental to trigonometry. This falls under the logical fallacy of circular thinking by trying to prove an idea with the idea itself.

“There are no trigonometric proofs because all the fundamental formulas of trigonometry themselves are based on the truth of the Pythagorean theorem,” mathematician Elisha Loomis had written in 1927.

But two American high school classmates, Ne’Kiya Jackson and Calcea Johnson, achieved the “impossible” during their senior year of high school in 2023.

Now they’ve published those results, along with nine more pieces of evidence.

“There were many times when we both wanted to give up on this project, but we decided to persevere and finish what we started,” Jackson and Johnson wrote in their paper.

The Pythagorean theorem describes the relationship between the three sides of a right triangle. It is incredibly useful for engineering and construction and has been used by humans for centuries before the equation was attributed to Pythagoras, including, some claim, in the building of Stonehenge.

The theorem is a fundamental law of trigonometry, which essentially calculates the relationships between sides and angles of triangles. You probably remember that you had the equation a^{2}+b^{2}= c^{2} drilled into you at school.

“Students may not realize that two competing versions of trigonometry are squeezed into the same terminology,” Jackson and Johnson explain.

“In that case, trying to make sense of trigonometry can be like trying to understand an image where two different images are printed on top of each other.”

By disentangling these two related but different variations, Jackson and Johnson were able to come up with new solutions using Sines’ Law, thus bypassing direct circular thinking.

Jackson and Johnson outline this method in their new paper, although they note that the boundary between trigonometric and non-trigonometric is somewhat subjective.

They also point out that according to their definition, two other experienced mathematicians, J. Zimba and N. Luzia, also proved the theorem using trigonometry, disproving previous claims that this was impossible.

In one of their proofs, the two students took the definition of calculating with triangles to the extreme by filling a larger triangle with sets of smaller triangles and using calculus to find the dimensions of the original triangles.

“It looks like nothing I’ve ever seen,” mathematician Álvaro Lozano-Robledo of the University of Connecticut told Nikk Ogasa at Science News.

All told, Jackson and Johnson provide one proof for right triangles with two equal sides and another four proofs for right triangles with unequal sides, leaving at least five more for “the interested reader to discover.”

“To have a paper published at such a young age, it’s really mind-boggling,” said Johnson, who is now studying environmental engineering. Jackson is now studying pharmacy.

“Their results draw attention to the promise of a fresh perspective on the field from students,” said Della Dumbaugh, eeditor-in-chief of the journal in which they were published.

This research has been published in the *American Mathematical Monthly*.