The latest mathematical news involves two high school students from New Orleans who have found a new way to prove the Pythagorean Theorem, which of course states that for a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Now of course there are many proofs of this theorem (the link above gives some), but, according to the Guardian article referenced below (click on screenshot), and the POCIT article below that (also click on screenshot), one form of proof is missing: a proof based on trigonometry. The Guardian article reports that this proof was supplied by Calcea Johnson and Ne-Kiya Jackson from St Mary’s Academy.
Here’s a tweet from the American Mathematical Society:
Click to read the Guardian article:
Now this is somewhat above my pay grade, but perhaps a mathematician will weigh in. The Guardian notes this:
The 2,000-year-old theorem established that the sum of the squares of a right triangle’s two shorter sides equals the square of the hypotenuse – the third, longest side opposite the shape’s right angle. Legions of schoolchildren have learned the notation summarizing the theorem in their geometry classes: a2+b2=c2.
As mentioned in the abstract of Johnson and Jackson’s 18 March mathematical society presentation, trigonometry – the study of triangles – depends on the theorem. 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.
Johnson and Jackson’s abstract adds that the book with the largest known collection of proofs for the theorem – Elisha Loomis’s The Pythagorean Proposition – “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’.”
What is the circularity? To me (and again I may be wrong) it’s instantiated in the section below from the Wikipedia article. The supposed trigonometric “proofs” of the Theorem depends on an identity:
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
The identity is
. . . and that is proven simply from the observation of right triangles and the definition of sines and cosines (see here). Thus, since this equation is derived from definitions and the use of a right triangle, it cannot itself be used to prove the theorem. Solid proofs have depended on arguments from geometry, rearranging similar triangles, algebra, and differentials. What is said to be new here is the proof of the Theorem using trigonometry, but not in a circular way. Because I haven’t seen the students’ proof, I can’t comment on it, and probably don’t have the chops, either. I’ve put one mathematician’s video interpretation below.
A lot of the press has gotten this mixed up, like this article below (click to read):
It says this:
The Pythagorean theorem is a fundamental theorem in trigonometry that describes the relationship between the three sides of a right-angled triangle. It is expressed with the formula a² + b² = c².
The theorem holds true in every plausible example and has been around since the days of the Ancient Greeks.
American mathematician Elisha Loomis argued that no mathematician has been able to establish its truth without using circular logic i.e. without using Pythagorean Theorem. However, others have argued it can be proved using the notion of similar triangles.
Well, that’s a bit misleading, as you can establish the truth of the Pythagorean theorem without using circular logic—so long as the proof is not a trigonometric one. And that’s what the teens are said to have overcome. However, I noticed this morning that an addendum had been added to the article.
This article was corrected on March 28, 2023. The article previously stated that the Pythagorean theorem had yet to be proven without using circular logic. The article has been amended to acknowledge that this argument was posited by American mathematician Elisha Loomis and is not accepted by all mathematicians.
Even so, the correction (which doesn’t itself invalidate what the students did) still leaves out the fact that even if Loomis was wrong, his claim was not that it couldn’t be proven without using circular logic, for it can be. What he said is that it can’t be proven trigonometrically without using circular logic.
Here’s a video which is one person’s interpretation of Johnson and Jackson’s proof based on the slides he saw of their presentation at the AMS meeting:
It’s quite clever, and doesn’t depend on the use of the Pythagorean Identity, merely the definition of “sine”. Note that the person who made this video isn’t 100% sure that he’s presenting Johnson and Jackson’s proof.
So, were they the first people to prove the Pythagorean theorem without trigonometry? I did some Googling, and found this at the site cut-the-knot.org (click to read), which apparently came out before 2019.
It merely reprises another trigonometric proof of the Pythagorean Theorem, one given in the paper below. This is from the article above:
Now, Jason Zimba has showed that the theorem can be derived from the subtraction formulas for sine and cosine without a recourse to sin²α + cos²α = 1. [JAC: the latter is of course the Pythagorean Trigonometric Identity.]
The paper it refers to is this:
J. Zimba, On the Possibility of Trigonometric Proofs of the Pythagorean Theorem, Forum Geometricorum, Volume 9 (2009) 275-278
And I found it online. You can read Zimba’s paper by clicking on the screenshot:
Zimba says he can derive the Pythagorean Trigonometric Identity independently of the Pythagorean Theorem, and thus you can use the former to prove the latter without circularity. He defines sine and cosine without using right angles, and then employs a subtraction procedure to come up with the Pythagorean Identity. That identity then establishes the truth of the Pythagorean Theorem. Here’s a bit of the ending, but it’s a short paper.
The author of the cut-the-knot.org piece says that Zimba’s proof is correct. But what do I know? If it is, then Johnson and Jackson were not the first people to produce a non-circular trigonometric proof of the Pythagorean theorem. But even if that’s the case, if the video gives an accurate take of the two students’ proof, what they did is still remarkably clever.
I suppose mathematicians will weigh in on the Internet in the next few days, so stay tuned.
25 thoughts on “A trigonometric proof of the Pythagorean theorem at last? The news said it was done by two high school students!”
So many trigonometric theorems and identities use UNIT circular logic…
Well, it has to be submitted, I suppose, but I did not know that a proof of the Gougu theorem was “missing”, that is intriguing – though I was aware of lots of proofs on the Wikipedia page (not that I could write them all down in a pinch).
Nice the detour to the Gougu identities – important and interesting!…
The title says “A non-trigonometric proof of the Pythagorean theorem”, but “they have proven Pythagoras’s theorem by using trigonometry. “.. so the “non” is a typo, ..?
Yes, I’ll fix it, thanks!
[ wipes brow ]
The proof in the YouTube video seems to be solid. In this proof you have to assume a<b so that the geometric series will converge; it’s a simple matter to prove the PT in the case of isosceles right triangles, where a=b. Zimba’s proof, as well, seems to be solid.
The proof in the YouTube video is a lovely proof, exquisite.
A terrific accomplishment by high school students!
But I think what’s going on is that in Euclidean geometry, everything is being driven by the Euclidean parallel postulate. In the familiar form given by John Playfair, the postulate states that through a given point, there is only one line parallel to a given line. It’s well known that the Pythagorean Theorem is equivalent to the Euclidean parallel postulate, and similarity–required for defining the sine function–also requires the Euclidean parallel postulate. So I think any proof requiring similarity, and I suspect the Law of Sines, is ultimately circular, using assumptions already known to be equivalent to the Theorem of Pythagorus.
Playfair, by the way, was a friend of the geologist James Hutton. He was present with Hutton when they discovered the famous angular unconformity at Siccar Point, which provided powerful evidence for deep geological time.
Yeah, I forgot to consider this little inconvenient fact about similarity. (You can define the sine and cosine of acute angles by using right triangles but it’s similarity that makes these definitions well-defined.) I guess I’ll have to go back and look at the proofs again (no worries, I’m retired.)
Coincidentally, it was the anniversary of Hutton’s death on Sunday (I only know because I noted below the line on the Hili Dialogue). I remember visiting Hutton’s Unconformity on the Isle of Arran during a geology field trip decades ago.
Sadly, I’m not in a position to comment on the mathematical proof – like our host said, it’s well above my paygrade.
The fact that the axioms of euclidean geometry imply the Pythagorean Theorem (PT) doesn’t make every proof of PT circular — it just makes PT a theorem. What’s needed here is that the proof doesn’t use other theorems which are derived from PT, and that does seem to be the case here.
Without the parallel postulate, PT may be false (like on the surface of a sphere) so there won’t be a proof of PT in Euclidean geometry that doesn’t ultimately depend on the parallel postulate.
The main ingredients of this proof are (1) basic facts about similar triangles which are needed, as you noted, to give meaningful definitions of sin and cos, and (2) the formula for cos(a-b), whose derivation here uses only basic facts about triangles (sum of angles is 180 and properties of similar triangles again).
I doubt that PT is needed for the basic facts about similar triangles (but I’m no expert), and the given derivation of the formula for cos(a-b) does not use the distance formula nor any other result which seems to derive from PT. So it seems right to me.
Every proof of the PT follows a basic template: Using the axioms of Euclidean Geometry, establish a finite chain of “if A, then B”implications, the last B being the Pythagorean Theorem, in which the PT is never used to prove any of these implications. Both the proofs in Dr Coyne’s original thread meet this standard, I feel safe in saying. It’s okay to use results involving similarity because these results can be obtained directly from the axioms, without recourse to the PT. (My own favorite proof of the PT uses the fact that the angle sum of a triangle is 180 degrees. This fact is equivalent to the PT, but it’s okay to use it to prove the PT because I can prove it directly from the axioms.) So the proofs, I think they’re solid.
Really, the issue in this case is what is meant by a trigonometric proof. Whatever one might conclude, it needs to be understood that the starting points are the axioms of Euclidean Geometry.
The circularity is broken because the parallel postulate* is assumed to be true in Euclid and PT is not assumed. You could define “Pythagorean Geometry” that uses PT instead of the parallel postulate but you then have to prove that PT is equivalent to PP to show that the two geometries are identical.
The same, by the way, applies to Playfair’s axiom. It is commonly quoted as a replacement for the PP but it still had to be proven to be equivalent.
Wikipedia lists a number of statements that are equivalent to the parallel postulate. The most bizarre IMO – and one I find difficult to believe – is
* The parallel postulate as given by Euclid
The upper limit to the area of a triangle seems bizarre. But what happens is this: If you assume all the axioms of Euclidean geometry except the parallel postulate, and instead assume there can be more than one line through a given point parallel (not intersecting) another line, you can prove there is an upper limit to the area of a triangle. The more-than-one-parallel postulate is the postulate for hyperbolic geometry, and in hyperbolic geometry, the angles in a triangle always add up to less than 180 degrees. And it turns out the area of a triangle will be proportional to its “angle defect” (how much the angle sum falls short of 180). Thus an upper bound for the area of a triangle. This all sounds strange, but it’s one of the greatest discoveries in mathematics that hyperbolic (and other non-Euclidean geometries) are exactly as valid as Euclidean geometry.
Likewise, in Riemannian (spherical) geometry where every pair of lines intersects eventually (because all lines are great circles around a sphere), the angles of a triangle always add up to more than 180 degrees, and the area is likewise proportional to the angle defect. Eventually you reach a triangle where all three angles are 180 degrees, and it’s no longer a triangle but a great circle.
Indeed. Areas of triangles have an upper bound in both spherical geometry and hyperbolic geometry. Apparently the fact that not-PP implies a finite upper bound in triangle area was proven by Gauss.
I’m just adding this hastily – perhaps I didn’t catch it :
The students’ proof (is the abstract verboten to print here? it is
short…) uses the “law of sines”, being “independent” of the Pythagorean identity :
“In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles.
a/sin(a) = b/sin(b) = c/sin(c) = 2R
[…] R is the radius of the triangle’s circumcircle.
if you don’t need the 2R part, the equation is quoted as :
sin(a)/a = sin(b)/b = sin(c)/c
… that’s all.
the Math Stack Exchange is a good website (and is not a forum) for answers – e.g. a search for “trigonometric proof theorem”, a sample post is below (though it has a -1 rating) :
.. and there is some material on the concern of “circular reasoning” (not circles per se!). A good guide is if there are answers, especially by highly ranked members, sometimes professors of mathematics.
Wouldn’t drawing the 2nd and 3rd lines used to begin the whole process technically be considered as circular logic because you’re introducing the idea of 2 nonexistent lines to continue with the process?
Why? There’s no reason why you can’t construct any lines you need, as long as you define them sufficiently
I noticed if I asked chat GPT for a non circular proof using trigonometry it had no issue giving me an answer. Maybe I just don’t know enough.
Could be because the Chat AI is taking information from recent news articles about the discovery and giving you that info
GPT-4’s information cuts off sometime in 2021 if I remember correctly.
ChatGPT’s current knowledge cutoff is September/October 2021.
Are you sure it gave a CORRECT answer?
Chat GPT is brilliant at providing logical-sounding explanations for any desired topic.
But half the time it’s completely lying.
For example, when asked to judge a legal decision, it sometimes makes up fake laws or fake court precedents. When asked to write a scientific research paper, it often makes up fake data and fake journal references. It could just as easily put fake justifications in a math proof.
Clickbait, mostly. For instance, even Scientific American says “The other trigonometric proofs of the theorem that have appeared in the past include a few that are described on mathematician Alexander Bogomolny’s website.”
They were not the first to develop a trigonometric proof of Pythagoras theorem. I am surprised Scientific American worded it this way.