New Perspectives on Pythagoras’ Theorem from High School Innovators

 A bonus math contest question has led to a remarkable breakthrough: two high school students have uncovered 10 novel proofs of Pythagoras’ theorem, a core mathematical principle that describes the relationship between the sides of right-angled triangles. Their achievement directly challenges the long-standing belief that using trigonometry to prove Pythagoras' theorem would be circular reasoning.

In 1927, mathematician Elisha Loomis argued that trigonometric proofs for Pythagoras' theorem were impossible, as all trigonometric principles rely on this very theorem. Nevertheless, in 2023, high school students Ne'Kiya Jackson and Calcea Johnson took on this mathematical challenge, persevering through obstacles to not only solve it but to publish nine additional proofs in a mathematics journal.

Pythagoras’ theorem, symbolized by the formula $a^2 + b^2 = c^2$, is essential to engineering, construction, and other fields. This fundamental law of trigonometry may even have been applied in ancient structures like Stonehenge.

Through their research, Jackson and Johnson noticed two subtly different approaches in trigonometry—one focusing on ratios of triangle sides, the other on angles. By distinguishing between these approaches, they avoided the circular reasoning that previously deterred mathematicians from proving the theorem through trigonometry.

Applying an adapted version of the Law of Sines, they found a new way to prove the theorem without traditional methods. They also noted that the boundary between “trigonometric” and “non-trigonometric” proofs may be more flexible than originally thought. By their definitions, previous mathematicians J. Zimba and N. Luzia also indirectly achieved proofs using trigonometric approaches.

In one creative proof, Jackson and Johnson calculated the side lengths of a large right triangle using smaller triangles, incorporating calculus to find measurements—a method University of Connecticut mathematician Álvaro Lozano-Robledo described as entirely unique.

Altogether, they presented one proof for right triangles with equal sides and four proofs for those with unequal sides, leaving five additional proofs as an exercise for the reader.

Ref: Jackson, N., & Johnson, C. (2024). Five or Ten New Proofs of the Pythagorean Theorem. The American Mathematical Monthly, 131(9), 739–752. https://doi.org/10.1080/00029890.2024.2370240.