THERE ARE NO LIMITS TO INTELLECTUAL CURIOSITY.

There are multiple ways to establish the Pythagorean Theorem ab initio.  At least one President of the United States is credited with a proof.  No less an authority than Rene Descartes, however, has cautioned us that the field of trigonometry uses Pythagoras as foundational, and so has mathematics education ever after.  "And since that particular field of study was discovered, mathematicians have maintained that any alleged proof of the Pythagorean theorem which uses trigonometry constitutes a logical fallacy known as circular reasoning, a term used when someone tries to validate an idea with the idea itself."  Put another way, without Pythagoras being true a priori, no trigonometry.  "Because the theorem underpins trigonometry, experts have thought it impossible to prove [using trigonometry], since you can't use trigonometry without already accepting that the theorem is correct."

Not so fast.

In the 2000 years since trigonometry was discovered it's always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis) the author 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.

The authors? Two seniors at St. Mary's Academy in New Orleans.  High school seniors.  "When the American Mathematical Society met in Georgia last week, [Calcea] Johnson and [Ne'Kiya] Jackson were the only high schoolers at the meeting, according to New Orleans television news station WWL."

We'll let the philosophers quibble over whether it is possible to speak of "opposite," "adjacent" and "hypotenuse" without implicitly creating a continuum of Pythagorean triples.  The high-schoolers were encouraged by participants in the academic conference to send their results out for peer review.  That's how it works, present your working paper at a conference, get some discussant comments, send it out for publication.

Good.  It is not yet time to abandon hope.  "Alluding to how St Mary’s slogan is 'No excellence without hard labor,' the two students credited their teachers at the all-girls school in New Orleans’s Plum Orchard neighborhood for challenging them to accomplish something which mathematicians thought was not possible."  Let us be grateful that the teachers at St. Mary, an institution apparently with some civil rights chops, didn't buy the nonsense about classical mathematics perpetuating privilege, and might be a high school properly preparing its graduates for college.