Calcea Johnson and Ne’Kiya Jackson are two high school students at St Mary’s Academy in New Orleans, Louisiana. They have managed to do what mathematicians have been unable since the theorem of right angled triangles was written in Shuba Sutra of Baudhayan, 2800 years ago in India: they have proven the “pythagorean” theorem of right triangles with trigonometry, something that was thought impossible.  It was considered circular reasoning (defining something in terms of itself) because trigonometry is predicated on right triangles.

(I will only call it the “pythagorean” theorem because that’s what most people know it as.  Also, I’m wrongly calling it a proof even though it hasn’t been peer reviewed yet.)

New Orleans teens’ Pythagorean proof gains compelling evidence

Compelling evidence supports the claims of two New Orleans high school seniors who say they have found a new way to prove Pythagoras’s theorem by using trigonometry, a respected mathematics professor said, even if the students’ “really important and fantastic” achievement is not the first time trigonometry has been used to prove the theory, as their school apparently touted.

Álvaro Lozano-Robledo, of the University of Connecticut, spoke this week in a series of TikTok videos, addressing international reports about Calcea Rujean Johnson and Ne’Kiya Jackson.

Johnson and Jackson, students at St Mary’s Academy, recently gave a presentation at a regional meeting of the American Mathematical Society outlining their discovery.

The 2,000-year-old Pythagorean theorem states that the sum of the squares of a right triangle’s two shorter sides is the same as the square of the hypotenuse, the third side opposite the right angle. The notation associated with the theorem – a2+b2=c2 – is something encountered in many a geometry class.

For generations, mathematicians maintained that any alleged proof of the Pythagorean theorem based in trigonometry would constitute a logical fallacy known as circular reasoning: seeking to validate an idea with the idea itself.

In the abstract for their 18 March talk in Atlanta, at an event that drew presenters from prominent universities, Johnson and Jackson noted that the book thought to hold the largest known collection of proofs for the theorem, The Pythagorean Proposition by Elisha Loomis, “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 Johnson and Jackson said they found a way to use the trigonometry law of sines to prove Pythagoras’s theory in a way “independent of the Pythagorean trig identity sin2x+cos2x=1” – without resorting to circular reasoning.

Scientific American published an article called 2 High School Students Prove Pythagorean Theorem. Here’s What That Means, explaining more detail than “news” sources could.

This video below explains how Johnson and Jackson proved their theorem, by scaling identical triangles and turning this into the sum of an infinite series.  It’s one of those obvious proofs that make you wonder why nobody thought of it before.