What started as a school project turned into a result presented at a major US maths conference: a new trigonometric proof of Pythagoras’ theorem, built without leaning on the very theorem it aims to show.
A 2,000-year-old cornerstone under fresh scrutiny
Pythagoras’ theorem is usually one of the first big ideas pupils meet in geometry. It links the three sides of a right-angled triangle.
For a triangle with sides a and b around the right angle, and hypotenuse c, the relationship is simple:
a² + b² = c² for every right-angled triangle in flat (Euclidean) geometry.
This statement sits behind a huge range of things: from basic carpentry and surveying to satellite navigation, 3D graphics and robotics.
Mathematicians have produced hundreds of proofs of the theorem over more than two millennia. Many are geometric, some are algebraic, some use areas or rearrangements. A few are beautiful enough to be taught just for their elegance.
Yet one thing was widely assumed: you cannot prove Pythagoras using trigonometry alone without sneaking the result in through the back door.
The long-standing trigonometry problem
Trigonometric functions like sine and cosine are tightly tied to right-angled triangles. At school, they are often defined using the lengths of the sides in such a triangle.
- sine of an angle = opposite side ÷ hypotenuse
- cosine of an angle = adjacent side ÷ hypotenuse
- tangent of an angle = opposite side ÷ adjacent side
From those ratios, one can quickly derive one of the most famous identities in mathematics:
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sin²(x) + cos²(x) = 1
Yet many standard arguments for that identity already rely, directly or indirectly, on Pythagoras’ theorem. That creates a logical loop: using Pythagoras to define trig, then using trig to prove Pythagoras.
For decades, teachers and mathematicians would tell curious students that a purely trigonometric proof of Pythagoras, free of that circular reasoning, just was not available.
Two teenagers who refused to accept “that’s impossible”
In 2022, US high school students Ne’Kiya Jackson and Calcea Johnson, then studying at St Mary’s Academy in New Orleans, decided to challenge that belief.
Over four years, they worked through angle relations, similar triangles and proportion arguments, trying to rebuild trigonometry on more basic geometric ground.
Their goal: define trigonometric functions without smuggling Pythagoras into the foundations, then use those functions to arrive at a² + b² = c².
They began with simple constructions: right-angled triangles inscribed in circles, repeated angle divisions, and families of similar triangles. From those, they established proportion relationships that did not depend on the theorem they were ultimately targeting.
Rebuilding sine and cosine from scratch
Using these constructions, Jackson and Johnson introduced sine and cosine not as “length ratios you memorise”, but as functions tied to angles and similarity.
Because similar triangles keep the same ratios of sides, they could argue that certain side-length relationships depended only on the angle, not on the size of the triangle. That step allows you to define sine and cosine purely from geometry and similarity.
Once that groundwork was in place, they showed that these functions satisfy relationships that simplify the geometry of right-angled triangles. That led them, through a chain of algebraic and geometric steps, to a form equivalent to:
sin²(x) + cos²(x) = 1, obtained without taking Pythagoras as a starting point.
From there, moving back to side lengths, they could translate that identity into the familiar statement a² + b² = c² for a right-angled triangle.
The key point is logical: Pythagoras no longer hides in the definitions. The theorem becomes a consequence, not an assumption.
From classroom project to national stage
By 2023, the pair were ready to show their work. They presented their results at the annual meeting of the Mathematical Association of America, held in Atlanta.
Seasoned mathematicians in the audience were intrigued. The problem of finding a non-circular trigonometric proof had been mentioned informally for years.
Their work was later accepted for publication in the journal American Mathematical Monthly, a sign that the argument held up under expert scrutiny.
Their paper actually contains more than one proof. One of their constructions can be adapted to generate several distinct proofs of Pythagoras, all staying within the trig-based framework they carefully set up.
Why this matters beyond one theorem
At first glance, this might sound like a clever party trick: proving an ancient theorem in a fresh way. In mathematics, fresh proofs often have much wider value.
New approaches can highlight forgotten structures, simplify complex reasoning, or suggest new problems. Sometimes, a different proof of a known result ends up powering better algorithms or more stable numerical methods.
By showing that core parts of trigonometry can be built without leaning on Pythagoras, Jackson and Johnson opened the door to alternative foundations for parts of geometry and analysis.
| Aspect | Traditional teaching | Jackson & Johnson approach |
|---|---|---|
| Starting point | Assume Pythagoras, define trig ratios from side lengths | Use angle properties and similar triangles to define trig |
| Main tool | Area formulas, algebraic manipulation | Trigonometric identities derived without Pythagoras |
| Logical structure | Risk of circular reasoning | Clear separation between assumptions and theorem |
| Outcome | Standard proof of a² + b² = c² | Multiple trig-based proofs of a² + b² = c² |
A signal to young people: your ideas count
Since their breakthrough, the two former classmates have moved into demanding university courses. Jackson is studying pharmacy at Xavier University of Louisiana, while Johnson is pursuing environmental engineering at Louisiana State University.
Both have said they want their work to send a message to younger students who might feel that high-level mathematics is out of reach.
Their story underlines that curiosity, persistence and a willingness to question “that’s just how it’s done” can matter as much as age or academic titles.
The pair spent years returning to the same diagrams and equations, testing for hidden assumptions or small logical gaps. That slow, careful checking process reflects what professional mathematicians do daily but is rarely visible in school textbooks.
Potential ripple effects in maths and beyond
Rebuilding a proof from different starting points can shift how subjects are taught. Some educators already see potential uses of the Jackson–Johnson approach in classrooms.
- It could provide a more logically clean route from geometry to trigonometry.
- It might help advanced pupils understand the idea of “foundations” in maths.
- It offers teachers a real story about teenagers pushing against a long-standing assumption.
In research fields that rely heavily on geometry and trigonometry – such as computer graphics, signal processing, geolocation, or even parts of machine learning – clearer foundations can reduce ambiguity in algorithms and models.
While this single result is unlikely to overhaul those areas, it encourages mathematicians to look again at other “settled” facts. Some of those re-examinations do eventually lead to cleaner methods or better numerical stability.
What “non-circular proof” really means
One technical phrase keeps coming up around this story: circular reasoning. In mathematics, a proof is circular when its steps rely, directly or indirectly, on the statement being proven.
For Pythagoras and trigonometry, the risk looks like this: define sine and cosine using right-angled triangles and Pythagoras, then derive an identity involving sine and cosine, then use that identity to “prove” Pythagoras again. Logically, nothing new has been shown.
A non-circular proof starts from assumptions independent of the target theorem and reaches the result without looping back to it.
Jackson and Johnson’s work is significant because they took that logical structure seriously. They checked that the facts they used to define trigonometric functions did not sneak Pythagoras in through geometry shortcuts.
Trying similar reasoning at home or in class
For pupils and teachers, this story can be a prompt for small experiments, even without access to the full research paper.
One simple exercise is to list everything usually assumed when defining sine and cosine in class. Which steps use right-angled triangles? Where does Pythagoras first appear? Could you instead use a circle and similar triangles to motivate the definitions?
Another activity is to compare several known proofs of Pythagoras – area rearrangements, algebraic proofs, proofs using similar triangles – and ask which assumptions each proof really needs. That habit of checking assumptions is at the heart of the Jackson–Johnson project.
Stories like this can also shift how students view mathematics. Rather than a fixed collection of rules, it becomes a living subject where old results can be looked at from new angles, and where even school students can, sometimes, move the conversation forward.