Two American teenagers shake 2,000 years of history with a fresh breakthrough on Pythagoras’ theorem: Even from a single school project sometimes, even the most experienced researchers find themselves at a standstill. One such research project was initiated in Louisiana and it proved to be so. Two adolescent students asked the question that was regarded by the majority of people as already answered. The result? An ancient textbook theory, the Pythagorean theorem, was revived with a new face.
It is not merely a story of a mathematical finding, but also a story of how inquisitiveness has no age. Whereas the adult thought, What is new to explore here? two schoolgirls thought, Why not re-explore it? And it is this thought that brought them to the field of mathematics studies.
The Theorem That Shaped Everything from School Math to Modern Technology
The Pythagorean theorem is a fundamental principle of geometry that almost every student has studied. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a² + b² = c²
This is not a formula, but the basis of numerous technologies in the contemporary world. It is used by Masons to measure 90 degree angles of walls. It is applied by engineers to compute distances and forces. Computer graphics, signal processing, GPS systems, all of them are based on this principle somehow.
Mathematicians have given hundreds of proofs of this theorem in more than two millennia; some involving geometry, some algebra, some cutting and recombining shapes. Thus, schools teach it as knowledge that has been established. And it appeared to the story to end there.
A “Forbidden” Path: Proof Using Only Trigonometry
Yet, there was one approach that people avoided—proving the Pythagorean theorem using only trigonometry. The problem was that trigonometric functions like sine and cosine are often derived from the Pythagorean theorem itself. For example, the identity:
sin²(x) + cos²(x) = 1
is often determined by the unit circle and the Pythagorean relationship. When the same name, then, is used to demonstration Pythagoras the fallacy is circular, that is to say, it is circular. Because of this reason, a good number of experts thought that a purely trigonometric proof could not exist.
Two Teenagers and a Big Question
In 2022, two students from St. Mary’s Academy in New Orleans—Ne’Kiya Jackson and Calcea Johnson—challenged this accepted truth. Their simple question was:
Is it truly impossible, or have people simply given up trying?
They approached it not as a competition project, but as genuine research. Under the guidance of their teachers, they worked for years. By early 2023, they had a groundbreaking result—a whole family of proofs of the Pythagorean theorem that use trigonometry, but do not presuppose Pythagoras anywhere.
Making Waves at a Math Conference
They also gave their work at a conference of the Mathematical Association of America (MAA) in March 2023. It was expected by many to be some sort of classroom anticap. What came out was a methodical, technical case which dealt with one issue that professional mathematicians had long shirked.
Experts later reviewed their work and the results were published in a prestigious journal American Mathematical Monthly. It is such an uncommon feat among high school children.
How Did They Avoid Circular Reasoning?
Their strategy was subtle. They did not define sine and cosine in the usual way (i.e., based on ratios derived from the unit circle or Pythagoras). Instead, they started with:
- The properties of angles
- Similar triangles
- Proportions
They defined trigonometric functions as ratios that remain constant in similar triangles. Then they proved the relationships between these ratios—without assuming a² + b² = c².
When they later derived identities like sin²(x) + cos²(x) = 1, it emerged from their own construction, not from a pre-existing assumption.
From Trigonometry Back to Pythagoras
Now that sine and cosine stood on their own, they established relationships between different right triangles. After a series of algebraic steps, the relationship between the sides condensed into the familiar form:
a² + b² = c²
The significance lies not just in the final formula, but in the architecture that leads to it—without borrowing the very thing being proven.
Why the Mathematical Community Was Impressed
This finding did not alter the Pythagoras theorem. It altered the premise on which we know it to be. It demonstrated that trigonometry does not merely rely on Pythagoras, but it can be used to substantiate it.
In mathematics, it is very vital to know the assumptions on which they are based. This type of change may affect the curricula, the way of teaching, and the culture of proof.
Potential Impact from the Classroom to Algorithms
Will this immediately change your phone? Probably not. But algorithms involving distance, angles, and measurements—graphics, robotics, navigation, AI—all rely on fundamental geometry.
Reorganizing concepts can sometimes lead to more stable calculations, better numerical methods, and new avenues for error control—especially when dealing with very large or very small numbers.
“Math is not for me”A Message for Thinkers
Interestingly, neither of the two students is pursuing a career in mathematics—one is studying pharmacy, the other environmental engineering. Yet they demonstrated that a PhD isn’t necessary to ask profound questions.
Their story shows that research sometimes begins with a question as simple as a homework problem—and that curiosity, if nurtured, can push the boundaries of knowledge.
Even Established Knowledge is an Open Framework
We tend to think of school theorems as museum-pieces–as dusty and fixed and final. However, in mathematics, they tend to be more like “workshop tools” – that can be disassembled, edited and re-assembled.
And the moral of this tale is:
Taken seriously, curiosity can open up new avenues in the most ancient truths.
FAQs
1. What is Pythagoras’ theorem?
It states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²).
2. What did the two students discover?
They created proofs of Pythagoras’ theorem using trigonometry without relying on the theorem itself, avoiding circular reasoning.
3. Why was a trig-based proof considered impossible?
Because most trigonometric formulas are built using Pythagoras, so using them to prove it usually leads to a logical loop.
4. Why is this discovery important?
It changes how mathematicians think about the foundations of geometry and trigonometry and may influence how these topics are taught.
5. Does this change the theorem itself?
No, the theorem remains the same; the discovery offers a new way to prove and understand it.
