Their work does not replace Pythagoras’ theorem, but it challenges how that ancient result can be proven, using tools once thought off‑limits for the job.
A 2,000-year-old cornerstone put back on the table
Pythagoras’ theorem is usually one of the first big results pupils meet in geometry. It links the three sides of any right-angled triangle.
If the shorter sides are called a and b, and the longest side opposite the right angle is c, then the theorem states:
a² + b² = c² for every right-angled triangle.
For more than two millennia, mathematicians have produced hundreds of proofs. Some rely on pure geometry, others use algebra, and even US President James Garfield famously gave one based on area calculations.
One method has always been considered off limits: using trigonometry – the study of angles, sines and cosines – to prove Pythagoras. Textbooks usually warn that trigonometry is built on this theorem, so any trig proof risks going round in circles.
Two teenagers tackle a forbidden proof
In 2022, two US high-school students, Ne’Kiya Jackson and Calcea Johnson, decided to challenge that unwritten rule. Studying at St. Mary’s Academy in New Orleans, they set themselves a bold question: can you prove Pythagoras using only trigonometry, without secretly assuming it?
For four years, they worked after class, testing constructions and checking every assumption. They had to rebuild parts of school mathematics from the ground up.
Their key idea: define trigonometric functions without ever using Pythagoras’ theorem, then use those functions to reach a² + b² = c².
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Rebuilding trigonometry from basic geometry
Standard lessons define sine and cosine using right-angled triangles and, quietly, Pythagoras’ theorem. Jackson and Johnson took a different route.
They started with basic Euclidean geometry:
- angles formed when lines cross
- similar triangles and proportional sides
- facts about circles and arcs
From these raw ingredients, they constructed right-angled triangles and other shapes where side ratios could be controlled purely by angle properties, not by Pythagoras.
Only then did they introduce the familiar functions sine and cosine as specific ratios between sides in these carefully built triangles.
Step by step, they showed that these functions obeyed relationships that do not depend on the theorem they wanted to prove. That avoided the circular logic that had blocked previous attempts.
The famous identity that changes everything
Once sine and cosine were defined geometrically, the students considered their most famous link:
sin²(x) + cos²(x) = 1
In school, that identity is usually derived from Pythagoras’ theorem. Jackson and Johnson instead reached it from their angle constructions and proportional reasoning.
With that identity in hand, they could relate side lengths in right-angled triangles purely through trigonometric expressions. After a sequence of algebraic manipulations, the familiar equation emerged again:
a² + b² = c², obtained without ever assuming Pythagoras along the way.
In other words, they used trigonometry that had been cleaned of any hidden reliance on Pythagoras, and then showed that this “independent” trigonometry leads straight back to the ancient theorem.
From school project to international spotlight
By 2023, the two students had several versions of their argument. One method, they say, can be adapted to generate five more distinct proofs, all trig-based and all non-circular.
They presented their results at the annual conference of the Mathematical Association of America (MAA) in Atlanta. For a pair of teenagers used to classroom blackboards, stepping onto a professional stage was a shock.
Seasoned mathematicians in the audience took notice, not because the theorem was new, but because the method was.
The work was later written up and accepted for publication in the journal American Mathematical Monthly, a widely read research and education outlet in mathematics. That move signalled that specialists had checked the reasoning and found it sound.
What makes their approach different
Mathematicians have long believed that any “trigonometric proof” of Pythagoras was doomed to smuggle the theorem back in through a side door. Jackson and Johnson’s work shows that this pessimism was too strong.
To understand why experts care, it helps to contrast three broad styles of proof for Pythagoras:
| Type of proof | Main tools | Typical feature |
|---|---|---|
| Geometric | Areas, rearrangements, similar triangles | Often visual, based on shapes and symmetry |
| Algebraic | Coordinates, equations, vectors | Uses x, y, and formulas on a number line or grid |
| Trigonometric (their work) | Angles, sine, cosine, identities | First rebuilds trig from basic geometry, then proves a² + b² = c² |
Their contribution slots into the third column: an approach that treats trigonometry not as a dependent tool, but as a standalone framework from which Pythagoras can be derived.
Two new voices in a very old conversation
Beyond the technical achievement, the story has a strong human side. Both students come from a historically Black Catholic school in New Orleans.
Calcea Johnson is now studying environmental engineering at Louisiana State University. She says the project shows that serious mathematics is not restricted to professors in distant universities.
“Even students can add to what we know,” she has said in interviews, stressing patience as much as talent.
Her classmate, Ne’Kiya Jackson, went on to study pharmacy at Xavier University of Louisiana. She often highlights persistence rather than raw brilliance as the key ingredient.
Their message to younger pupils is simple: tough problems are not owned by adults. Long, careful work in a school setting can still make people at international conferences stop and listen.
Possible ripple effects for future maths
Pythagoras’ theorem sits quietly behind many technologies: 3D graphics, GPS, engineering design, and basic distance calculations in data analysis. Any new angle on that theorem invites fresh questions.
Mathematicians are already asking where this kind of foundational rethinking could lead. If trigonometry can be built in a different order, other parts of school mathematics might be reorganised too.
That might one day yield:
- new classroom sequences that introduce angles and distances differently
- alternative proofs for other classic results, such as cosine rules or circle theorems
- cleaner assumptions for algorithms in geometry-heavy fields like robotics or computer vision
The link to artificial intelligence is indirect but real. Many machine-learning techniques work with high-dimensional spaces, where distance and angle play central roles. A sharper understanding of geometric foundations can sometimes lead to more stable, efficient numerical methods.
Key notions worth unpacking
What is a non-circular proof?
A proof is called circular when it secretly assumes the very fact it aims to establish. In the context of Pythagoras, that might mean defining sine and cosine using the a² + b² = c² relation, then using those functions to “prove” the same relation again.
Jackson and Johnson’s main technical achievement lies in stripping such hidden assumptions away. They rely only on basic properties agreed upon before Pythagoras appears: how angles behave, how similar triangles work, and how ratios can be compared.
Why thousands of proofs still matter
At first glance, adding yet another proof to a hefty collection might feel redundant. But mathematicians value different proofs for distinct reasons.
- Some are easier to teach to beginners.
- Some shed light on connections to other topics, like algebra or complex numbers.
- Some are easier to adapt to new problems or generalisations.
Trig-based proofs of Pythagoras that avoid circular reasoning sit neatly in the second and third categories. They help clarify how trigonometry hangs together and may hint at broader structures in geometry that have not yet been fully mapped.
How a teenager could try a similar project
For pupils inspired by this story, the path does not require exotic equipment. It does require time, curiosity and a willingness to be stuck for weeks.
One realistic starting point is to choose a familiar formula – not necessarily as famous as Pythagoras – and ask: “Which steps in my textbook proof are assumed rather than explained?” From there, students can attempt to rebuild the argument using only the most basic properties they fully understand.
Teachers can turn this into extended projects or maths club activities. Pupils might, for instance, try to prove the area formula for a circle using only properties of polygons, or derive the tangent addition formula from geometric constructions rather than memorised identities.
The risk is frustration: many attempts simply will not work. The benefit, as Jackson and Johnson’s story shows, is that occasionally a stubborn question from a school desk can nudge a 2,000‑year‑old pillar of mathematics in a slightly new direction.
Originally posted 2026-03-12 11:12:42.