High school students may have just discovered an 'impossible' proof to the 2,000yearold Pythagorean theorem
Two high school seniors have presented their proof of the Pythagorean theorem using trigonometry — which mathematicians thought to be impossible — at an American Mathematical Society meeting.
Two high school students say they’ve proved the Pythagorean theorem using trigonometry — a feat mathematicians thought was impossible.
While the proof still needs to be scrutinized by mathematicians, it would constitute an impressive finding if true.
Calcea Johnson and Ne'Kiya Jackson, who are seniors at St. Mary's Academy in New Orleans, presented their findings March 18 at the American Mathematical Society’s (AMS) Spring Southeastern Sectional Meeting.
"It's an unparalleled feeling, honestly, because there's nothing like it — being able to do something that people don't think that young people can do," 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."
Pythagoras' 2,000yearold theorem, which states that the sum of the squares of a right triangle's two shorter sides is equal to the square of the hypotenuse, is the basis for trigonometry. Trigonometry, which comes from the Greek words for triangle ("trigonon") and to measure ("metron"), lays out how the side lengths and angles in a triangle are related, so mathematicians thought that using trigonometry to prove the theorem would always include some hidden expression of the theorem itself. Thus, proving the theorem with trigonometry would constitute a failure of logic known as circular reasoning.
Remarkably, Johnson and Jackson say they can prove the theorem without using the theorem itself. Because the findings have not yet been accepted into a peerreviewed journal, however, it's still too soon to say whether their proof will ultimately hold up.
Related: Mathematicians make rare breakthrough on notoriously tricky 'Ramsey number' problem
In their abstract, Johnson and Jackson quote from a 1927 book by the American mathematician Elisha Loomis (1852 to 1940) called "The Pythagorean Proposition," which contains the largest known collection of proofs of the theorem — 371 solutions, according to research published in the Journal of Targeting, Measurement and Analysis for Marketing. "There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem,” Loomis wrote.
But "that isn't quite true," the teenagers wrote in the abstract. "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 sin2x+cos2x=1." In other words, the high school seniors said they can prove the theorem by using trigonometry and without circular reasoning.
"It is unusual for high school students to present at an AMS Sectional Meeting," Scott Turner, director of communications at AMS, told Live Science in a prepared statement.
Despite their young age, the AMS has encouraged the high schoolers to submit their findings to a scientific journal. "Following their conference presentation, their next step would be to look into submitting their work to a peerreviewed journal, where members of our community can examine their results to determine whether their proof is a correct contribution to the mathematics literature," Catherine Roberts, executive director at AMS, said in the statement.
Johnson and Jackson's achievement has not gone unnoticed in math circles. "We celebrate these early career mathematicians for sharing their work with the wider mathematics community and we encourage them to continue their studies in mathematics," Roberts added.
Live Science newsletter
Stay up to date on the latest science news by signing up for our Essentials newsletter.
Sascha is a U.K.based trainee staff writer at Live Science. She holds a bachelor’s degree in biology from the University of Southampton in England and a master’s degree in science communication from Imperial College London. Her work has appeared in The Guardian and the health website Zoe. Besides writing, she enjoys playing tennis, breadmaking and browsing secondhand shops for hidden gems.

Yury Schwarzman Unfortunately, the mathematical typesetting for the trigonometric formula used in the text has been lost: the identity reads cos^2x + sin^2x=1 (that is, cosine squared and sine squared, rather than cosine and sine of the doubled angle, respectively).Reply 
George Kosinski It's weird how logic  combined with numerous examples  tells us the Pythagorean theorem must be true for every right angle, and yet, at the same time, it's so hard to prove that it is.Reply 
ClueAnon Simple HTML tags in a comment maybe won't make it through, but it should look like this: cos<sup>2</sup>x + sin<sup>2</sup>x = 1Reply
I hate to rain on the parade, but this is just a trivial restatement of the theorem. 
laudahwiz
Read more carefully  they said their proof works independently of the equation you quoted.ClueAnon said:Simple HTML tags in a comment maybe won't make it through, but it should look like this: cos<sup>2</sup>x + sin<sup>2</sup>x = 1
I hate to rain on the parade, but this is just a trivial restatement of the theorem. 
jayaram1961 They have used infinite series solution. I have a much simpler and better trigonometry proof without infinite series solution.Reply