High School Students Discover Unique Proofs of Pythagoras’ Theorem

In a remarkable achievement, high school students Calcea Johnson and Ne’Kiya Jackson have made waves in the mathematical community with their innovative proof of Pythagoras’ theorem through trigonometry. Their groundbreaking work emerged from a local high school competition in 2022, culminating in both students receiving keys to the city of New Orleans and even personal recognition from former First Lady Michelle Obama.

Their findings, which have been published in the journal American Mathematical Monthly, shed new light on the age-old theorem represented by the equation a² + b² = c². This principle allows for the calculation of the length of any side of a right-angled triangle provided the lengths of the other two sides are known. Traditionally, the theorem has been demonstrated via algebraic and geometric methods, but proving it through trigonometric identities has been deemed nearly impossible due to the reliance of trigonometric principles on the assumption that Pythagoras’ theorem is valid—creating a potential loop of circular reasoning.

Despite these challenges, both Johnson and Jackson managed to independently uncover trigonometric proofs of the theorem, accomplishing what only a select few professional mathematicians have achieved. Their collaboration led them to present their findings at the American Mathematical Society’s regional meeting in Atlanta in March 2023. Following positive feedback, they submitted their research for final peer review and publication.

Their study introduces five new methods of proving the theorem using trigonometry, along with an additional methodology that leads to five more proofs, bringing the total to ten. Remarkably, nine of these proofs are entirely novel, as only one had been previously acknowledged at the conference.

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

“To have a paper published at such a young age—it’s really mind-blowing,” adds Calcea Johnson.

Johnson expressed her enthusiasm for the rising interest in STEM fields, stating, “It’s very exciting for me because I know when I was growing up, STEM wasn’t really a cool thing. The fact that so many people are now interested in STEM and mathematics truly warms my heart.”

In their paper, the authors highlight a common source of confusion among high school students when learning trigonometry, which stems from two distinct interpretations of the subject using identical terminology. This overlap can muddle understanding, akin to viewing a picture with two images layered one over the other. By isolating and focusing on one interpretation of trigonometry, they argue, a wealth of new proofs for Pythagoras’ theorem becomes accessible.

Currently, Ne’Kiya Jackson is pursuing a doctoral degree in pharmacy at Xavier University of Louisiana, while Calcea Johnson is studying environmental engineering at LSU’s Roger Hadfield Ogden Honors College. Johnson takes pride in being a role model for young women and women of color in STEM disciplines, emphasizing the importance of representation in these fields.

Della Dumbaugh, the editor-in-chief of American Mathematical Monthly, praised the duo’s work, noting, “The Monthly is honored and delighted to publish the work of these two students. Their results call attention to the promise of the fresh perspective of students on the field. They also highlight the important role of teachers and schools in advancing the next generation of mathematicians.”

“This work echoes the spirit of Benjamin Finkel when he founded the Monthly in 1894 to feature mathematics within reach of teachers and students of mathematics,” Dumbaugh added.

For further information on their study, refer to the article titled Five or Ten New Proofs of the Pythagorean Theorem, published in the American Mathematical Monthly (2024). DOI: 10.1080/00029890.2024.2370240.