Two teens created groundbreaking trigonometric proofs of the Pythagorean Theorem

This is an updated version of a story first published on Nov. 1, 2024

For centuries, students have learned that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Known as the Pythagorean Theorem, this formula—expressed as a² + b² = c²—has more than 300 known proofs. Most of them use algebra or geometry. But until recently, almost no one thought it could be proven with trigonometry, since that branch of math depends heavily on the theorem itself.

That assumption fell apart when two high school seniors from St. Mary’s Academy in New Orleans, Ne’Kiya Jackson and Calcea Johnson, decided to take on a challenge that seemed impossible. What started as a school contest grew into a historic moment in mathematics.

Every December, St. Mary’s Academy holds a math competition for its students. In 2022, the bonus problem asked for something audacious: create an original proof of the Pythagorean Theorem using trigonometry. The task promised $500 to the winning entry, but it had long been assumed that such a proof couldn’t be done.

Ne’Kiya and Calcea were not math prodigies, but they were determined. They spent two months filling notebooks and throwing away page after page of calculations. Calcea’s mother recalled the trash can overflowing with paper. Ne’Kiya’s mother admitted she had no idea what her daughter was working on. Their teacher, Michelle Blouin Williams, didn’t expect the problem to be solved, but she encouraged creativity.

Despite frustration and self-doubt, the two seniors pressed on. By early spring, each had discovered something extraordinary.

Calcea’s proof became known as the “Waffle Cone.” She began with a right triangle, labeled its angles alpha and beta, then stacked identical copies in an infinite pattern. The repeating shapes formed a figure resembling a waffle cone, and through trigonometric relationships, she reached the Pythagorean result.

Ne’Kiya’s proof used a different approach. She placed a right triangle inside a circle, drew a perpendicular bisector, and divided the figure into smaller right triangles. By analyzing those relationships, she also reached the same formula.

Both methods were entirely original. Before their work, the only known trigonometric proof came from mathematician Jason Zimba in 2009. That meant these high school students had doubled the number of such proofs in existence.

Their teacher submitted the discoveries to the American Mathematical Society conference in March 2023, where Ne’Kiya presented. The audience response was overwhelming. Soon, news spread worldwide. Their achievement appeared in international media, earned praise from the governor of Louisiana, and even drew a congratulatory message from former First Lady Michelle Obama. New Orleans celebrated them with keys to the city.

But for Ne’Kiya and Calcea, the spotlight was complicated. Both recognized that their race and gender shaped people’s surprise. As young African American women in a field where they are underrepresented, they were keenly aware that some saw their accomplishment as unlikely. They wanted the work itself—the originality of their proofs—to stand at the center of the story.

The environment at St. Mary’s Academy played a crucial role. Founded just after the Civil War by an African American nun, the school has long pushed its students to aim high. Strict rules on dress, phones, and conduct are balanced by a culture of belief and support. For 17 years, the school has posted a 100% graduation rate and 100% college acceptance rate.

Students described the experience as both demanding and uplifting, with a strong sense of sisterhood. Many attend with the help of scholarships, yet all face high expectations. For Ne’Kiya and Calcea, that atmosphere gave them confidence to pursue a problem others had written off.

When they graduated, every member of their class went to college with scholarship aid. Ne’Kiya accepted a full ride to Xavier University to study pharmacy, while Calcea, the class valedictorian, chose environmental engineering at Louisiana State University.

The story did not end at the math contest. After peer review, their paper was published in The American Mathematical Monthly in 2024. The journal described it as an honor to feature the work of such young scholars.

The published article included more than their first discoveries. Together, they outlined five complete proofs and hinted at five more, bringing the total to ten, nine of which are entirely new. In their writing, they also proposed a teaching insight. They argued that trigonometry actually has two overlapping forms, and that separating them could reduce confusion for students. This distinction, they suggested, helped unlock new proofs of the theorem.

For the young mathematicians, the publication felt surreal. “I didn’t think it would go this far,” Ne’Kiya admitted. Calcea called the experience “mind-blowing,” especially knowing how much interest in math and STEM their work had sparked.

Their achievement carried weight beyond math. As role models, they inspired younger students—especially girls and students of color—to see themselves in science and math careers. “We are both able to be such a positive influence in showing that young women and women of color can do these things,” Calcea said.

Mathematics leaders also highlighted the broader lesson. Della Dumbaugh, editor-in-chief of The American Mathematical Monthly, noted that the results “highlight the important role of teachers and schools in advancing the next generation of mathematicians.” She stressed how students’ fresh perspectives can reinvigorate even ancient fields.

While many of their classmates earned scholarships to college, Ne'Kiya received a full ride to Xavier University’s College of Pharmacy in New Orleans, and Calcea, the valedictorian, went on to study environmental engineering at Louisiana State University. Though neither plans to pursue a career in math their work has already influenced the discipline. They continue to explore proofs on their own, but they are also busy with demanding college programs.

For them, the journey proved something larger: talent flourishes when given belief and opportunity. What began as a simple contest question became a reminder that excellence is not bound by age, race, or gender.

The proofs by Ne’Kiya Jackson and Calcea Johnson expand the mathematical community’s understanding of the Pythagorean Theorem. Their insight into the structure of trigonometry offers a new way to teach the subject, potentially making it less confusing for students worldwide.

By demonstrating that long-assumed boundaries in mathematics can still be crossed, their work encourages other young people to explore fresh approaches in science.

Their achievement shows that new ideas can come from anyone, at any stage of education, and reinforces the importance of supportive teachers and schools in unlocking hidden potential.

Note: The article above provided above by The Brighter Side of News.

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