Pythagorean theorem on a chalkboard. (Credit: Viktoriia_M/Shutterstock)
NEW ORLEANS — A high school math contest has turned into a history-making event thanks to a pair of young girls. These two teenage math stars have developed five new proofs of the Pythagorean theorem, one of mathematics’ most fundamental principles.
Their work, now published in The American Mathematical Monthly, challenges long-held beliefs about how this cornerstone of geometry can be proven. Ne’Kiya Jackson and Calcea Johnson, who completed this work while still in high school, have accomplished something that has eluded mathematicians for centuries: creating entirely new ways to prove the famous a² + b² = c² equation. Their achievement is particularly remarkable given that hundreds of proofs of the Pythagorean theorem already exist, making new discoveries increasingly rare.
“I was pretty surprised to be published” says Jackson in a statement. “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 Johnson. “It’s very exciting for me, because I know when I was growing up, STEM [science, technology, engineering, and math] wasn’t really a cool thing. 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.”
The project began when both students independently tackled a bonus question in their high school math contest that offered a $500 prize for creating a new proof of the Pythagorean theorem. What started as a contest entry evolved into months of rigorous mathematical research, ultimately leading to their presentation at the American Mathematical Society’s Southeastern Sectional conference in March 2023 — making them the youngest presenters in attendance.
Their paper not only presents five new proofs but also provides a systematic method for discovering additional proofs, potentially opening the door for even more mathematical discoveries. The students’ work is particularly significant because it focuses on trigonometric proofs, which some mathematicians had previously claimed were impossible.
This achievement is even more impressive considering that Johnson, who was valedictorian of her class at St. Mary’s Academy High School, and Jackson, now studying pharmacy at Xavier University of Louisiana, completed this sophisticated mathematical research while balancing their regular high school coursework. Their success challenges traditional assumptions about who can contribute to advanced mathematics and when such contributions can be made.
“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, and to let other young women know that they are able to do whatever they want to do. So that makes me very proud to be able to be in that position,” says Johnson.
Paper Summary
Methodology
The students developed their proofs by examining how new right triangles could be created from an original right triangle. They focused specifically on triangles whose angles were combinations of the original triangle’s angles. This methodical approach led them to discover that they could consistently create new triangles with specific angle measurements, which became the foundation for their proofs. Their systematic method involved creating these new triangles in various ways and then using established geometric principles to prove the Pythagorean relationship.
Key Results
The research produced five complete, novel proofs of the Pythagorean theorem, with the potential for at least five more using their developed method. Each proof uses different geometric constructions and trigonometric relationships to demonstrate that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. Four of their proofs work for non-isosceles right triangles, while the fifth proof works for all right triangles.
Study Limitations
The first four proofs presented in the paper have a notable limitation: they don’t work for isosceles right triangles (triangles where both non-right angles are 45 degrees). Only their fifth proof works for all right triangles. Additionally, their method for discovering new proofs is limited to creating triangles with angles that are specific combinations of the original triangle’s angles.
Discussion & Takeaways
The paper challenges the long-held belief that trigonometric proofs of the Pythagorean theorem are impossible or circular. It also provides a systematic method for discovering new proofs, suggesting that more discoveries are possible. Perhaps most importantly, this work demonstrates that significant mathematical discoveries can come from unexpected sources, including high school students. The authors’ success story highlights the importance of mathematical competitions and mentorship in fostering young talent in mathematics.
Funding & Disclosures
The paper acknowledges several key individuals who supported the research, including Mr. Rich, a volunteer math teacher at St. Mary’s Academy, Professor Lawrence Smolinsky of Louisiana State University, and Dr. Leslie Meadows of Georgia State University. The authors declared no potential conflicts of interest. The research appears to have been conducted independently without external funding, having originated from a high school math contest offering a $500 prize for a new proof of the Pythagorean theorem.
