The result of their after-school obsession is a new way of proving one of the most famous rules in mathematics – a result that has stunned experts, sparked debate, and reminded everyone that fresh ideas can come from unexpected places.
A 2,000-year-old theorem meets Gen Z
Pythagoras’ theorem is one of the first serious pieces of maths many of us meet at school. It states that, in 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: a² + b² = c².
The idea goes back at least 2,000 years and probably further. Dozens of proofs are known: geometric, algebraic, even some using rearrangements that can be shown with cardboard cut-outs. Textbooks present it as settled territory.
What nobody had seen before was a proof that relies purely on trigonometry – the branch of maths that studies angles and their relationships with lengths – without quietly smuggling Pythagoras’ theorem back in through the back door.
For decades, mathematicians suspected a fully trigonometric proof of Pythagoras’ theorem was impossible without circular reasoning.
That long-held belief has now been challenged by two young women who, when they began, were still in secondary school.
Who are the teenagers behind the breakthrough?
In 2022, US high-school students Ne’Kiya Jackson and Calcea Johnson, from Louisiana, unveiled a new family of proofs of Pythagoras’ theorem. Their work emerged from several years of independent study and guidance from teachers, carried out alongside the usual pressures of homework, exams, and university applications.
Jackson later went on to study pharmacy at Xavier University of Louisiana. Johnson enrolled in environmental engineering at Louisiana State University. Neither comes from a traditional elite maths research track, which makes their story resonate even more strongly with students who rarely see themselves in academic headlines.
“Even students can contribute to the progress of knowledge,” Johnson has said, framing their work as a message to younger pupils as much as to professional mathematicians.
➡️ Poop From Young Donors Reverses Age-Related Decline in The Guts of Older Mice : ScienceAlert
➡️ Battle lines drawn as U.S. accelerates construction of second Ford-class aircraft carrier USS Kennedy while USS Ford heads for combat, deepening global rifts and igniting a fierce debate over power, security, and the price of military dominance
➡️ Why emotional processing takes longer than logical reasoning
➡️ From February 15, hedges taller than 2 meters and planted less than 50 cm from a neighbor’s property must be trimmed or homeowners could face penalties
➡️ Why placing a bowl of coffee grounds near radiators is trending and what it really does for indoor air quality
➡️ I bought a tiny basil seedling for R$ 1.57 – and it took over my backyard
➡️ Sheets shouldn’t be changed monthly or every two weeks: an expert reveals the precise frequency
➡️ Over 60 and feeling less adaptable? The brain is actually conserving energy
Why a trigonometric proof is such a big deal
On the surface, trigonometry and Pythagoras’ theorem are tightly linked. In school, the definitions of sine and cosine usually rely on right-angled triangles and, explicitly or not, on the relationship a² + b² = c². That creates a logical loop: if your definition of trig functions already assumes Pythagoras, you can’t use those same functions to prove the theorem without going in circles.
That loop fuelled a common belief in maths education: a purely trigonometric proof of Pythagoras, free from any hidden use of the theorem itself, was not possible. Jackson and Johnson set out to challenge that assumption.
How they side-stepped the circular reasoning trap
The key move was to start from geometry, not from the standard school definition of trigonometric functions. The pair built right-angled triangles and related geometric figures using only basic Euclidean tools: straight lines, angles, and proportions between segments.
From there, they defined sine and cosine in a more fundamental way, anchored in angle relationships and proportional lengths, rather than in the usual “opposite over hypotenuse” recipe that smuggles Pythagoras in from the start.
Once those functions were defined, they could be manipulated using algebra and angle relationships to produce identities. One central identity in trigonometry is:
sin²(x) + cos²(x) = 1
By carefully building this identity from geometric principles instead of assuming it, they could then translate it back into statements about the sides of a right-angled triangle. Step by step, that path leads to a familiar-looking equation: a² + b² = c².
The result: a proof of Pythagoras’ theorem that never relies on the theorem itself, and never quietly assumes the usual trig rules that depend on it.
Multiple proofs, not just one
Jackson and Johnson did not stop at a single argument. In their paper, they presented several distinct trigonometric proofs, one of which serves as a kind of generator for further demonstrations. From that method, at least five additional proofs of the theorem can be constructed.
- One approach uses nested right-angled triangles and angle halving.
- Another relies on cleverly chosen similar triangles and proportional sides.
- A broader scheme allows families of new proofs to emerge from a single geometric configuration.
This richness impressed mathematicians because it suggests a deeper structure behind their work: they did not just find a mathematical curiosity, but opened a route toward an entire landscape of alternative arguments.
How the maths community reacted
After four years of effort, the two students presented their findings at the Mathematical Association of America’s annual meeting in Atlanta in March 2023. Their talk, tucked into a packed conference programme, quickly spread through word-of-mouth and social media.
Their work was later accepted for publication in the American Mathematical Monthly, a respected peer-reviewed journal.
Publication in that journal matters because it signals that experts have checked the logic carefully. Reviewers look not only for correctness, but also for originality and clarity. The fact that the paper passed this scrutiny gave their result weight far beyond the headlines.
While professional mathematicians do not see the theorem itself changing, many welcome the new proofs as fresh tools for teaching, outreach, and further research.
Why this matters beyond one theorem
The immediate consequence is conceptual: the work shows that even classical results can still admit new viewpoints. For researchers, that is a reminder that familiar territory can hide uncharted paths.
There are also potential long-term uses. Geometry and trigonometry underpin a wide range of applied fields. More flexible ways of thinking about triangles and angles can trickle into algorithms and models, sometimes in indirect ways.
Possible areas influenced by new geometric thinking
| Field | Connection to Pythagoras and trigonometry |
|---|---|
| Computer graphics | 3D rendering and animation rely on triangle meshes, distances, and angles. |
| Signal processing | Waveforms and oscillations use trigonometric functions at their core. |
| Engineering | Structural analysis, forces, and stress depend on geometric decompositions. |
| Artificial intelligence | Vector norms and distance-based learning methods use Pythagorean-type formulas. |
While Jackson and Johnson’s proofs do not instantly rewrite these subjects, they add to a toolbox that can inspire new algorithms and alternative formulations, especially in areas that rely on precise distance and angle calculations.
A signal to young people who think maths is “finished”
For many students, maths appears as a completed monument: everything settled, all the theorems already proved by distant geniuses. This story cuts against that perception. Two teenagers, working patiently with ideas taught in secondary school, managed to produce something professionals had not done before.
The message is less about genius and more about persistence, careful reasoning, and the willingness to question assumptions.
Teachers can use this case to show that asking, “Does it really have to be done this way?” can be a starting point for genuine research. A class discussing Pythagoras’ theorem might now compare traditional geometric proofs with the idea of a trigonometric approach, then ask what assumptions sit under each method.
Clearing up a few key terms
What is Euclidean geometry?
Euclidean geometry is the classic geometry taught at school: flat surfaces, straight lines, and familiar shapes like triangles and circles. It is based on axioms formalised by the ancient Greek mathematician Euclid. Jackson and Johnson’s work sits squarely in this framework, using its rules in creative new ways.
What is trig, really?
Trigonometry studies how angles and lengths relate, especially in triangles and periodic phenomena. Functions like sine and cosine link an angle to a ratio of sides or, in more advanced treatments, to positions on a circle. These functions appear everywhere from satellite navigation to audio compression.
By rebuilding parts of trigonometry from elementary geometry and then using that to prove Pythagoras’ theorem, the two students essentially rewired a small chunk of the logical architecture behind what many of us learn in school.
How students and educators might build on this
A practical classroom activity could involve asking pupils to list every assumption behind the “opposite over hypotenuse” definition of sine and cosine. From there, a teacher can guide them toward alternative ways of thinking: defining trig functions using circles or coordinate geometry, then comparing which definitions depend on Pythagoras.
Another angle is project-based learning: groups of students attempt to construct their own new proof of Pythagoras, using only tools they already understand. The goal is not to beat professional mathematicians but to experience, in miniature, the process of questioning a long-standing method and seeking a different route.
Stories like Jackson and Johnson’s show how such experiments sometimes grow beyond the classroom. Once in a while, a school project ends up changing how experts think about a theorem that has stood for two millennia.