Their work does not replace Pythagoras’ theorem, but it challenges long-standing assumptions about how this ancient result can be proven, using mathematical tools that were once considered inappropriate for the task.
A two-thousand-year-old foundation reconsidered
Pythagoras’ theorem is often among the first major ideas students encounter in geometry. It describes the relationship between the three sides of any right-angled triangle.
If the two shorter sides are labelled a and b, and the longest side opposite the right angle is c, the theorem states that a² + b² = c² for every right-angled triangle.
Over more than two millennia, mathematicians have devised hundreds of proofs. Some rely purely on geometry, others use algebra, and even US President James Garfield once presented a proof based on area calculations.
One approach, however, has long been considered forbidden: using trigonometry. Because trigonometric functions are usually introduced using Pythagoras’ theorem, any attempt to prove it with trigonometry has been seen as logically circular.
Two teenagers challenge an unwritten rule
In 2022, two US high-school students, Ne’Kiya Jackson and Calcea Johnson, decided to question that assumption. While studying at St. Mary’s Academy in New Orleans, they asked a bold question: could Pythagoras’ theorem be proven using only trigonometry, without relying on it at any hidden step?
For four years, they worked beyond regular classes, testing geometric constructions and scrutinising every assumption. Much of their effort involved rebuilding parts of school mathematics from first principles.
Their central insight was to define trigonometric functions without invoking Pythagoras’ theorem, and only then use those functions to derive the familiar equation a² + b² = c².
- How repeating the same setup reduces daily stress
- Greenland crisis: Denmark orders officials to switch off Bluetooth amid espionage fears
- Behavioral scientists say people who walk faster than average share consistent personality indicators
- Meteorologists confirm the jet stream will realign unusually early this February
- How many countries are there in the world?
- The world’s longest high-speed underwater train is now underway
- Day will turn to night: astronomers confirm the longest solar eclipse of the century
- Hygiene after 60: experts reveal the shower frequency that best supports long-term health
Reconstructing trigonometry from basic geometry
In standard lessons, sine and cosine are defined using right-angled triangles, with Pythagoras’ theorem quietly built in. Jackson and Johnson deliberately avoided this route.
They began with elementary Euclidean geometry: how angles behave when lines intersect, how similar triangles create proportional sides, and how circles and arcs are related.
Using only these elements, they constructed triangles in which side ratios depended solely on angle properties. Only after that did they introduce sine and cosine as specific ratios between sides in these carefully designed figures.
They then demonstrated that these functions obeyed key relationships without any dependence on Pythagoras’ theorem, thereby avoiding the circular logic that had previously blocked similar attempts.
The identity that unlocks the theorem
Once sine and cosine were defined geometrically, the students turned to their most familiar connection: sin²(x) + cos²(x) = 1.
In most classrooms, this identity is derived directly from Pythagoras’ theorem. Jackson and Johnson instead reached it through angle constructions and proportional reasoning alone.
With this identity established independently, they expressed the side lengths of right-angled triangles using trigonometric terms. After a series of algebraic steps, the classic result reappeared: a² + b² = c², achieved without assuming the theorem at any point.
In effect, they showed that trigonometry stripped of hidden assumptions naturally leads back to Pythagoras’ ancient result.
From classroom work to international attention
By 2023, the students had developed several versions of their argument. One approach, they note, can be adapted to produce five additional, distinct trigonometric proofs, all avoiding circular reasoning.
They presented their findings at the annual conference of the Mathematical Association of America in Atlanta. For two teenagers accustomed to school classrooms, presenting before professional mathematicians was an unexpected leap.
The audience took notice, not because the theorem itself was new, but because the method of proof was.
The work was later accepted for publication in the journal American Mathematical Monthly, indicating that experts had reviewed the reasoning and found it mathematically sound.
Why their method stands apart
For decades, many mathematicians believed that any trigonometric proof of Pythagoras’ theorem would inevitably reintroduce the theorem through hidden assumptions. Jackson and Johnson’s work demonstrates that this belief was overly restrictive.
To see why the result matters, it helps to compare three broad proof styles:
- Geometric proofs: rely on areas, rearrangements, and visual symmetry
- Algebraic proofs: use coordinates, equations, and vector methods
- Trigonometric proofs: rebuild trigonometry from basic geometry, then derive a² + b² = c²
Their contribution fits squarely into the third category, treating trigonometry as an independent framework rather than a tool dependent on Pythagoras’ theorem.
New voices in an ancient debate
Beyond the mathematics, the story carries a human dimension. Both students attended a historically Black Catholic school in New Orleans.
Calcea Johnson now studies environmental engineering at Louisiana State University, emphasising that meaningful mathematical work is not limited to established academics.
Ne’Kiya Jackson, who went on to study pharmacy at Xavier University of Louisiana, often highlights persistence rather than innate brilliance as the decisive factor.
Together, their experience reinforces a simple message: difficult problems are not reserved for professionals, and careful work in a school setting can still command global attention.
Potential implications for future mathematics
Pythagoras’ theorem underpins many modern technologies, from GPS systems to 3D graphics and engineering design. Any new perspective on such a foundational result invites further exploration.
Researchers are already considering whether similar foundational re-orderings could reshape other areas of school mathematics. Possible outcomes include:
- alternative ways to introduce angles and distances in classrooms
- new proofs for classic results such as cosine rules
- clearer assumptions for geometry-based algorithms in robotics and computer vision
Although indirect, these ideas also touch fields like artificial intelligence, where distance and angle play crucial roles in high-dimensional data spaces.
Key ideas worth clarifying
What defines a non-circular proof
A proof is circular if it quietly assumes the very statement it aims to establish. In the case of Pythagoras, this can happen when sine and cosine are defined using the theorem and then used to prove it.
Jackson and Johnson avoided this by relying only on properties introduced before Pythagoras appears: angle behaviour, similar triangles, and proportional reasoning.
Why additional proofs still matter
Even with hundreds of existing proofs, new ones serve important purposes. Some are easier to teach, others reveal deeper connections between topics, and some adapt more readily to new problems.
Non-circular trigonometric proofs belong to the latter categories, clarifying how trigonometry fits together and hinting at broader geometric structures.
How students might attempt similar projects
For students inspired by this example, no specialised equipment is required, but time, curiosity, and persistence are essential.
A practical starting point is to take a familiar formula and ask which steps in its standard proof are assumed rather than fully justified. From there, students can attempt to rebuild the argument using only basic, well-understood principles.
Such projects can be used in maths clubs or extended coursework. While many attempts may fail, Jackson and Johnson’s experience shows that a stubborn question from a school desk can still reshape how an ancient theorem is understood.
If you want to avoid loneliness at 70 and beyond, it’s time to say goodbye to these 9 habits
