Have two US teens found proof for 2,000-year-old Pythagoras theorem?

Two New Orleans high school students have stunned mathematicians after claiming they have proven Pythagoras’s theorem by using trigonometry - something academics have long thought to be impossible.

Pythagoras’ discovery that the sum of the squares of a right angled triangle’s two shorter sides equals the square of the hypotenuse – the third, longest side opposite the shape’s right angle, has been used for millennia. Generations of schoolchildren have learned the notation summarising the 2,000 year old theorem as a2+b2=c2.

Mathematicians have long argued the theorem, which uses trigonometry - the study of triangles - is impossible to prove, since you can't use trigonometry without already accepting the theorem is correct.

Calcea Johnson and Ne’Kiya Jackson, who are students of St Mary’s Academy, have flipped this assumption on its head, revealing their discovery of a new proof for the theorem in exciting new findings.

In the abstract of their mathematical society presentation, the students say that Elisha Loomis’s The Pythagorean Proposition, the largest known collection of proofs for the theorem: “flatly states that 'there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem’.”

"But that isn’t quite true: in our lecture we present a new proof of Pythagoras’s Theorem which is based on a fundamental result in trigonometry - the Law of Sines - and we show that the proof is independent of the Pythagorean trig identity \sin^2x + \cos^2x = 1."

In simpler terms, the high school seniors say they can prove the theorem using trigonometry and without using circular reasoning, when a person attempts to validate an idea with the idea itself.

“It is an unparalleled feeling honestly because there is just nothing like it - being able to do something that people don't think that young people can do," Ms Johnson told the New Orleans television news station WWL.

“You don’t see kids like us doing this – it’s usually, like, you have to be an adult to do this.”

The students recently gave a presentation of their findings at the American Mathematical Society south-eastern chapter’s semi-annual conference in Georgia - and, unsurprisingly, were the only high school students in the room.

The meeting was reportedly attended by math researchers from institutions such as the universities of Alabama, Louisiana State and Texas Tech.

Catherine Roberts, executive director for the American Mathematical Society, was reported to have encouraged the students to consider having their work examined by a peer-reviewed journal. “Members of our community can examine their results to determine whether their proof is a correct contribution to the mathematics literature,” Ms Roberts was quoted as having said.

Ms Johnson and Ms Jackson are set to graduate from St Mary's academy this spring and plan to embark on careers in environmental engineering and biochemistry.

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