Pythagorean theorem

Two US teenagers surprise with proof

(Bild: APA Pool/dpa/Julian Stratenschulte)

Two young female scientists with an affinity for mathematics have succeeded in doing what was long considered almost impossible by experts: they have proved the famous Pythagorean theorem (a² + b² = c²) using trigonometry - and they have done so several times over.

The crux of the matter: trigonometry is a branch of geometry, and its basic formulas are based on the assumption that Pythagoras' theorem is true. There is therefore a risk of so-called circular reasoning - i.e. a proof in which what is to be proven is already a presupposition.

According to the publisher, professional mathematicians have only succeeded twice in proving this without circular reasoning. In addition, there are hundreds of other proofs of the millennia-old theorem from other mathematical disciplines such as algebra.

Most people know it from school
The Pythagorean theorem is probably one of the few formulas that many people still have somewhere in the back of their minds from their school lessons. It is about the relationship between the side lengths of a right-angled triangle: The sum of the squares of the cathets (a and b) adjacent to the right angle is equal to the square of the hypotenuse (c), which is opposite the 90-degree angle. You can therefore calculate the length of any side of a right-angled triangle if you know the length of the other two sides.

Ne'Kiya Jackson (left) and Calcea Johnson were praised by former First Lady Michelle Obama.

(Bild: APA/dpa/Calcea Johnson)

In their recently published work, Calcea Johnson and Ne'Kiya Jackson have presented five ways of proving the theorem using trigonometry. There is also a method that enables five further proofs. Roughly speaking, they formed various new right-angled triangles with specific angle dimensions from a right-angled triangle ABC.

Mathematics competition as an impetus
The two came up with the idea as students in 2022 during a math competition at their high school in the USA. One of the questions was to create a new proof of the Pythagorean theorem. "500 dollars in prize money motivated us to take on this task," they write.

"The task turned out to be much more difficult than we had initially imagined, and we spent many late nights trying and failing to produce a proof," report Johnson and Jackson, whose work was published in the journal "American Mathematical Monthly".

Praise from former First Lady Obama
The teenagers sacrificed their free time for the project for several months, even working on it during their vacations. "There were many moments when we both wanted to give up on the project, but we decided to persevere and finish what we had started." In the end, according to the publisher, there was even praise from former First Lady Michelle Obama in addition to awards.

"I'm very proud that we were both able to make such a positive impact," explained co-author Johnson. The two authors had shown that young women are capable of doing this and "let other young women know that they can do anything they want to do", said Obama.

Johnson is now studying environmental engineering at Louisiana State University, Jackson pharmacy at Xavier University of Louisiana.