What started as a bonus question in a high school math competition has led to 10 amazing new ways to prove the ancient math rule of the Pythagorean Theorem. It has long been argued that it is impossible to use trigonometry to prove a theorem that is actually fundamental to trigonometry. This falls into the logical fallacy of circular thinking, which tries to prove an idea by the idea itself.

“There is no trigonometric proof, because all the fundamental formulas of trigonometry are based on the truth of the Pythagorean theorem,” mathematician Elisha Loomis wrote in 1927.

But two American classmates, Ne’Kiah Jackson and Calsea Johnson, achieved the “impossible” in their senior year in 2023. They have now published these results along with nine other proofs.

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

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

The theorem is a fundamental law in the field of trigonometry that essentially calculates the relationship between the sides and angles of triangles. You probably remember being taught the equation a in school²+b² =c².

“Students may not realize that two competing versions of trigonometry are written with the same terminology,” Jackson and Johnson explain. “Trying to understand trigonometry in this case can be like trying to understand a painting with two different images printed on top of each other.”

By separating these two related but different variations, Jackson and Johnson were able to find new solutions using the law of sines, avoiding direct circular thinking.

Jackson and Johnson describe this method in their new paper, but note that the line between trigonometric and non-trigonometric is somewhat subjective. They also note that two other accomplished mathematicians by definition, J. Zimba and N. Luzia, also proved the theorem using trigonometry, challenging past claims that it was impossible.

In one of their proofs, two students took the definition of triangle calculus to its extreme by filling in a larger triangle with a series of smaller triangles and using calculus to find the side measures of the original triangle.

“It’s like I’ve never seen anything like this,” University of Connecticut mathematician Alvaro Lozano-Robledo told Science News’ Nikku Ogasi.

Together, Jackson and Johnson provide one proof for right triangles with equal sides and four more proofs for right triangles with unequal sides, leaving at least five more proofs for the “interested reader.”

“It’s truly heartbreaking to have a paper published at such a young age,” says Johnson, who is currently studying environmental engineering. Jackson is currently studying pharmacy.

“The results highlight the promise of a new undergraduate perspective on the field,” says Della Dumbo, editor-in-chief of the journal in which they were published. This study was published on: American Mathematics Quarterly.