In a quiet US high school, two teenagers set out to tackle a maths problem that had supposedly been solved forever.
What began as a school-level research project has now travelled from a classroom in Louisiana to the pages of a major mathematical journal, forcing experts to revisit a theorem that has underpinned geometry for over two millennia.
A 2,000-year-old pillar of geometry
Pythagoras’ theorem is one of the first serious formulas many pupils meet in school. It governs right-angled triangles, the ones that have a perfect 90-degree corner.
In its classic form, the theorem states that for a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Written as an equation, that becomes:
a² + b² = c², where c is the hypotenuse and a and b are the other sides.
This idea appears in ancient Babylonian clay tablets and was formalised in Greek geometry. Since then, mathematicians have found hundreds of ways to prove it using geometry, algebra, even rearranging cut-out shapes.
One thing was considered off-limits: a proof based solely on trigonometry, the branch of maths built on angles and functions such as sine and cosine. The reason was logical. Standard definitions of sine and cosine already rely on Pythagoras’ theorem, so using them to prove it looks like going in circles.
Two teenagers, one forbidden proof
In 2022, US high school students Ne’Kiya Jackson and Calcea Johnson, then at St Mary’s Academy in New Orleans, decided to challenge that long-standing belief.
Over several years of work, they developed what many mathematicians thought could not exist: a proof of Pythagoras’ theorem that uses trigonometry, but does not assume the theorem is true in the first place.
They built trigonometry from the ground up, then used it to return to the very theorem that usually defines it.
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The key was to start not with sine and cosine as pre-packaged formulas, but with basic geometric facts: properties of angles, similar triangles and proportional sides. From those building blocks, they defined trigonometric functions in a way that stands independently from Pythagoras’ theorem.
How their trig-based proof actually works
Building triangles before building formulas
Jackson and Johnson began with carefully constructed right-angled triangles and other geometric figures. They relied on a classic idea: if two triangles have the same angles, their sides stand in the same ratios. That relationship between angles and side lengths is the foundation of trigonometry.
From there, they assigned values to ratios of sides for a given angle, which became their definitions of sine and cosine. At this stage, they avoided shortcuts that would smuggle Pythagoras’ theorem back into the argument.
- Angles and similar triangles set the stage.
- Ratios of sides for fixed angles define trigonometric functions.
- Algebraic manipulation of these ratios creates key identities.
- Those identities lead back to the familiar a² + b² = c².
The famous identity at the heart of the proof
One central relationship in trigonometry is often written as:
sin²(x) + cos²(x) = 1
In many textbooks, this identity comes directly from Pythagoras’ theorem applied to a unit circle. Jackson and Johnson reached it in the opposite direction: through angle properties and carefully justified ratios.
Once that identity was in place, they could connect the abstract “1” in the equation to the length of a hypotenuse relative to the other sides. With a series of algebraic steps, they showed that this identity forces the lengths in a right-angled triangle to obey the Pythagorean relationship.
The result: a proof of the theorem using trigonometric functions, but without a single step that assumes the theorem itself.
Their work shows that trigonometry does not have to sit on top of Pythagoras’ theorem; it can stand beside it and then lead back to it.
From a school project to a respected journal
Jackson and Johnson spent four years refining their ideas, checking for circular reasoning and building extra versions of the argument. In March 2023, they presented their findings at the annual conference of the Mathematical Association of America in Atlanta.
Their presentation turned heads. Experienced mathematicians, used to proofs from career academics, saw two teenagers calmly walking through arguments that punctured a long-held assumption.
Later, their work was accepted for publication in the American Mathematical Monthly, a well-known peer-reviewed journal. That gave their results a level of validation far beyond social media buzz.
Their paper does not stop at a single proof: one of their methods can be adapted to produce five additional, distinct proofs.
For professional mathematicians, this bundle of proofs is especially appealing. Multiple approaches often reveal hidden connections and new ways to generalise a result.
Why mathematicians care about “old” theorems
At first glance, proving Pythagoras again might sound like reinventing the wheel. The theorem has been proved in hundreds of ways; US President James Garfield even published his own proof in the 19th century.
Yet new proofs can shift how related areas of maths fit together. A trigonometric proof that genuinely stands apart from Pythagoras-based definitions nudges the foundations of geometry and analysis.
Some researchers see potential consequences for pure and applied maths alike. Fresh ways of relating angles and lengths can inspire new algorithms in computer graphics, optimisation, or even aspects of machine learning that rely heavily on geometric reasoning.
Possible future applications
While their result is theoretical, several areas could eventually benefit:
- Geometric algorithms: alternative formulations of distance could lead to more efficient computations in 3D modelling or robotics.
- Signal processing: trigonometric identities lie under the hood of many compression and filtering methods; new perspectives sometimes yield faster transforms.
- Artificial intelligence: many AI models operate in high-dimensional spaces where distance, angles and projections guide learning.
Role models who look like current students
Beyond technical impact, the story of Jackson and Johnson is reshaping how young people, especially girls and students of colour, see advanced maths.
Both have now moved into demanding university courses. Calcea Johnson is studying environmental engineering at Louisiana State University. Ne’Kiya Jackson is pursuing pharmacy at Xavier University of Louisiana.
They argue that serious mathematical research is not reserved for grey-haired professors; determined students can push the field forward too.
They have been explicit about wanting their journey to encourage others. They talk about passion and patience rather than genius. Four years of effort, revisions and setbacks sit behind their “overnight” success.
Key concepts worth unpacking
What exactly counts as circular reasoning?
Circular reasoning appears when a proof quietly assumes the result it claims to show. In the context of Pythagoras’ theorem, many textbook definitions of sine and cosine use a right-angled triangle and rely on the Pythagorean relationship to argue that sin² + cos² = 1.
If you then try to prove Pythagoras’ theorem using that identity, you end up depending on the theorem itself. The circle closes, and the proof loses logical independence. Jackson and Johnson’s care in avoiding this loop is what makes their work stand out.
How could a teacher bring this story into class?
This breakthrough can become more than a headline; it can turn into a teaching tool. A secondary school teacher might, for example:
- Ask pupils to collect different traditional proofs of Pythagoras’ theorem and compare their strategies.
- Use the story of the two students to start a project on similar triangles and ratios.
- Challenge classes to define simple trigonometric ideas from geometry, before presenting the standard formulas.
Such exercises show that mathematics is not a list of finished answers but a set of creative routes between ideas.
Why this matters beyond mathematics
Stories like this can change how society talks about scientific talent. When two teenagers, working from school labs and home desks, manage to publish in a respected journal, it sends a message about who is allowed to participate in high-level thinking.
Their work also reminds researchers that even the most familiar results can hide new questions. Established theorems remain fertile ground. With enough persistence, someone with fresh eyes can still reshape the conversation, just as Jackson and Johnson have done with a triangle drawn more than 2,000 years ago.
Originally posted 2026-02-12 13:06:32.