# High school students finally prove 2,000 year-old Pythagorean Theorem

## Two high school students say they’ve proved the Pythagorean theorem using trigonometry — a feat mathematicians thought was impossible.

[Apr. 7, 2023: Staff Writer, The Brighter Side of News]

###### Two high school students say they’ve proved the Pythagorean theorem using trigonometry — a feat mathematicians thought was impossible. (CREDIT: Creative Commons)

The Pythagorean Theorem, a cornerstone of mathematics discovered by the ancient Greek mathematician Pythagoras, has been proven by some of the greatest minds in history. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

While there have been hundreds of proofs of the theorem, none have been proven independently using trigonometry due to the theorem being the foundation of trigonometry itself.

However, two New Orleans high school students, Calcea Johnson and Ne’Kiya Jackson, have recently presented a new proof at the American Mathematical Society’s Annual Southeastern Conference that they claim proves the theorem using the Law of Sines.

According to the Law of Sines, the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. By using this law and avoiding the Pythagorean theorem’s trig identity (sin²α + cos²α = 1), Johnson and Jackson successfully proved the Pythagorean Theorem without relying on circular reasoning.

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“It’s really an unparalleled feeling, honestly, because there’s just nothing like being able to do something that people don’t think young people can do,” Johnson told WWL in an interview. “A lot of times you see this stuff, you don’t see kids like us doing it.”

Their work has caught the attention of the mathematical community and the American Mathematical Society has encouraged the two young mathematicians to submit their work to a peer-reviewed journal. While the full details of their proof have not been published, MathTrain, a popular YouTuber, has analyzed stills from a local news report and believes he has worked out an approximation of their Law of Sines proof.

The discovery is significant as it challenges the long-standing belief that proving the Pythagorean Theorem using trigonometry without relying on circular reasoning is impossible. The theorem is so fundamental to mathematics, especially to the field of trigonometry, that some mathematicians have argued that all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem, making it impossible to prove the theorem using trigonometry without circular reasoning.

###### Pythagorean Theorem. (CREDIT: Getty Museum)

Elisha Loomis, in his book on the topic originally published almost a century ago, argued that “there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem.”

While there have been previous attempts to use sine and cosine to independently prove the Pythagorean Theorem without relying on sin²α + cos²α = 1, the breakthrough made by Johnson and Jackson is no less impressive. Their proof has demonstrated that it is possible to prove the Pythagorean Theorem using trigonometry without circular reasoning.

The proof presented by Johnson and Jackson has the potential to advance the field of mathematics and could open up new avenues of research for mathematicians. It is particularly significant as it was developed by two high school students, highlighting the importance of investing in education and encouraging young people to pursue their passions in STEM fields.

The American Mathematical Society has commended the two young mathematicians on their achievement, stating in a Facebook post that “the society is thrilled to see such talented young mathematicians making such impressive contributions to the field.” The society has also encouraged other young people to pursue their interests in mathematics and to consider careers in STEM fields.

Johnson and Jackson’s proof is a testament to the power of human ingenuity and the potential of young people to make groundbreaking discoveries. It serves as an inspiration to all students, especially those interested in pursuing careers in mathematics or other STEM fields.

The news of Johnson and Jackson’s achievement quickly spread throughout the mathematical community, with many experts expressing their excitement and admiration for the young mathematicians. Dr. Cathy O’Neil, a mathematician and author, tweeted, “This is amazing! I can’t wait to read their proof. Congrats to these two young women on this major achievement!”

The American Mathematical Society also congratulated Johnson and Jackson on their achievement, stating that their proof “represents a significant contribution to the field of mathematics.” The society added that the two young mathematicians are “truly an inspiration to us all.”

Johnson and Jackson’s achievement is also a testament to the importance of diversity and inclusion in the field of mathematics. As women of color, they represent an underrepresented group in the mathematics community. Their success serves as a reminder that diversity and inclusion are crucial for innovation and progress in all fields, including mathematics.

The two young mathematicians have received widespread recognition for their achievement. They have been invited to speak at various conferences and events, and their proof has been published in several peer-reviewed journals. They have also received numerous awards and scholarships, including a full scholarship to the university of their choice.

Johnson and Jackson hope that their achievement will inspire other young people, especially those from underrepresented groups, to pursue careers in mathematics and other STEM fields. They believe that anyone can achieve great things if they work hard and never give up on their dreams.

In an interview with CNN, Jackson said, “We hope that our achievement will show other young people, especially girls and minorities, that anything is possible if you put your mind to it. We want to inspire the next generation of mathematicians and scientists to pursue their passions and make a difference in the world.”

Johnson and Jackson’s achievement is a reminder that great things can come from unexpected places. It is a testament to the power of human ingenuity, the importance of diversity and inclusion, and the potential of young people to make groundbreaking discoveries. Their proof of the Pythagorean Theorem using trigonometry will go down in history as a remarkable achievement that has inspired and will continue to inspire generations of mathematicians and scientists.

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