Right-Angles and Wrong Angles

It hit the news last week, that two high school kids had come up with a new proof of Pythagoras’s Theorem:

Two New Orleans high school seniors who say they have proven Pythagoras’s theorem by using trigonometry – which academics for two millennia have thought to be impossible – are being encouraged by a prominent US mathematical research organization to submit their work to a peer-reviewed journal.

I had decided to leave it alone. I had figured out enough of the story, that the two kids had done something very cool, which had then been way over-egged by the predictably clueless, Homerically lazy media. I’ve done these stories, and they can be tiring and tricky.

My friend Grant Cairns,* however, tipped it over the line, by pointing out an MAV tweet:

Good ‘ol MAV.

How a right-angled triangle is any more about “the world around us” than the number 3, God only knows. Grant summarised it as the MAV being “totally grounded in reality”. More accurately, the MAV is totally grounded by reality. To an earthbound hammer, everything is an earthbound nail.

OK, I’ve had my fun and now I must pay the price. I have to write about this sweet-absurd story.

There are, of course, and depending how you count, hundreds if not thousands of proofs of Pythagoras’s theorem. See, for example, the very long list at the brilliant cut-the-knot. Although many of these proofs can be regarded as minor variations of each other, there are still tens, probably hundreds, of fundamentally different proofs. The question is, is there a proof via trigonometry?

It is easy enough to come up with circular trig “proofs” (no pun intended). For example, we could start with the cosine rule:

    \[\boldsymbol{c^2 = a^2 + b^2 - 2ab\cos C\,.}\]

Then put C = 90 and there’s your Pythagoras. The problem of course, the circularity, is that Pythagoras is first required to prove the cosine rule. More directly, we can easily get Pythagoras via the fundamental trig identity

    \[\boldsymbol{\sin^2\theta + \cos^2\theta = 1\,.}\]

That’s so circular, it’s basically a dot.

With the fundamental trig identity in mind, it seems plausible that any trig proof of Pythagoras must of necessity be circular. That is what Loomis bluntly declared in his famous book of Pythagoras proofs (p viii).

But, there is more, and more prior, to trigonometry than the fundamental identity. The definitions of sin and cos, for example, come down to capturing the concept of similarity in right-angled triangles, and no Pythagoras is required for that. Similarly (still no pun intended), the proofs of the double angle formulas require no Pythagoras. So, one can at least contemplate coming up with a trig proof of Pythagoras.

Calcea Johnson and Ne’Kiya Jackson, two high school students, claimed to have done exactly that. Johnson and Jackson presented their circularity-free trig proof of Pythagoras at a regional meeting of the AMS,** after which it all hit the news.

Johnson and Jackson’s proof has not been publicly presented in print, meaning plenty of people had to puzzle unnecessarily,*** but their proof appears to be correct, free of circles, and very clever. The key diagram, as reconstructed by Grant, is as follows:

In brief, the original right-angled triangle ABC is used to create a larger right-angled triangle ABD. Then, the lengths of BD and AB are calculated using geometric series, their ratio is related back to the original triangle using the sin rule and the double angle formula for sin, and Pythagoras basically pops out. Readers can enjoy working out the details, or here is a very nice presentation and discussion by MathTrain:

Johnson and Jackson’s proof is complicated but elegant. It is very nice stuff, and of course it is great for school kids to come up with this. But, there is a but.

It’ll come as no great surprise to readers here that Johnson and Jackson’s is not the first trig proof of Pythagoras. This is discussed at some length at cut-the-knot, where a 2009 trig proof by Jason Zimba is outlined. Zimba’s very careful and well-written paper is also available online, courtesy of a kind redditer.

More generally, one has to ask what it means for a proof of Pythagoras to be a “trigonometry proof”, which is also discussed at cut-the-knot. For example, Grant has suggested that any proof of Pythagoras incorporating similarity, of which there are many, might be properly called a trigonometry proof. So, it is reasonable if not very generous to suggest that Johnson and Jackson were not the first to prove something that is not a thing.

But of course what what Johnson and Jackson did is very cool, and this post is not about them. This post is about the reporters, for whom, as always with these stories, there are lessons. In brief:

1) If you’re planning on reporting on a “new proof” then first ask a mathematician.

2) If you’re too lazy to find a mathematician then first check cut-the-knot.

3) Once you realise you’re too lazy to read cut-the-knot, ask a mathematician.

Three simple rules. Which almost no reporter will ever follow.

 

*) The greatest Head of Mathematics in the history of LaTrobe University.

**) Kudos to AMS for giving Johnson and Jackson the platform.

***) Non-kudos to AMS for not assisting Johnson and Jackson in getting their proof clearly visible in a timely manner.

23 Replies to “Right-Angles and Wrong Angles”

  1. Yeah. Their proof is very cool and it is really awesome that they had the chance to present it.

    The media coverage is annoying and deserves the whack.

    On the math side, (in first year) I like to start with series, then power series, then define some special series, which will become special functions, from which their properties can be divined, including e.g. identities that yield Pythagoras. This is not a good idea in high school of course but at university I like this approach because (a) it is new, (b) it is not circular, and (c) it leans into analysis and proofs.

    In high school there are a lot of really nice possibilities. For example, I like using the right-angled triangle to make a square. Probably it is a good idea to expose students to a few proofs. I’m not sure that actually happens though.

    1. The only “proof” of Pythagoras’ theorem that I remember being taught in high school was a trigonometry proof, indeed the

          \[\boldsymbol{\sin^2\theta + \cos^2\theta = 1\,.}\]

      proof.

      “On the math side, (in first year) I like to start with series, then power series, then define some special series, which will become special functions, from which their properties can be divined, including e.g. identities that yield Pythagoras. ”

      I don’t really think (I could be wrong, feel free to prove me so) that you can actually (feasibly) prove Pythagoras’ theorem using that method. Pythagoras’ theorem is a theorem about the plane, and unless you go through the hassle of actually precisely defining the plane (in a way that doesn’t assume Pythagoras theorem, so no metric spaces, or you’ll have to be much more creative with how you use them), you’re not actually going to be able to connect the abstract world of real(complex?) analysis to the (similarly) abstract world of plane geometry.

      1. Geez. Someone really did the dot proof of Pythag for you?

        I’m not sure what Glen is referring to, but you can talk about the plane and coordinates etc without throwing in distance.

        1. I was mistaken, they taught us the fifth of the false proofs here: https://www.cut-the-knot.org/pythagoras/FalseProofs.shtml. I confused the two because of how circular they are.

          It was the first or one of the first “vector proofs” that we did.

          Just to add a bit more to the cut-the-knot’s analysis, the motivation for the conventional dot product is Pythagoras’ theorem, so going backwards is not only applying the results in one framework to another (as cut-the-knot points out), but also fundamentally absurd.

          “I’m not sure what Glen is referring to, but you can talk about the plane and coordinates etc without throwing in distance.”

          This is true, which surprised me: I tried my hand at a proof of Pythagoras’ theorem using purely analytic geometry (without distances), and with a few assumptions: that the area of a parallelogram is its base times its height, and that a square with a side from (0, 0) to (a, b) will have a side from (0, 0) to (-b, a) or (b, -a), the proof of Pythagoras’ theorem is pretty natural.

      2. One absolutely can prove Pythagoras in that manner, and I have done so for many years. The plane can be simply defined as \mathbb{R}\times\mathbb{R}. The unit circle is the set (\sin x, \cos x) where x\in\mathbb{R}. There are a few more ingredients (to define a right-angled triangle, and so on) but nothing so severe as metric spaces.

        In any case, as I mentioned Pythagoras is not the main game in that sequence, rather the topics of sequences and series (especially power series, Cauchy product formula, exchanging limits (differentiating series), and infinity in general). But Pythagoras is a nice narrative, or at least a nice remark to end with, so long as you don’t focus on it too much.

        The metric spaces version of Pythagoras is good to remark on in the following year when looking again at analysis in a later subject, as that is a convenient setting for many general fundamental facts. Unfortunately I do not think we treat classical planar geometry as a whole subject, but that might be a natural example in subjects that deal with general algebra and groups.

        I guess the only reason I mentioned it was because some students had remarked to me that this is the first time they have seen a lecturer/teacher prove Pythagoras in class. Of course as soon as I heard that I shared with them my favourite Pythagoras proof (the square one mentioned in other places).

        1. The definition of the unit circle is the set of points of distance one from (0, 0) in the plane. I’m sorry, but I simply don’t see how you can define a unit circle without a notion of distance.

          Can you post a full explanation of how you prove Pythagoras in this manner?

          1. That Pythagoras follows from elementary analysis in the way I’ve indicated does not feel particularly important, original, or new. I think I’ve been explicit enough and it is a side issue that I don’t want to take too much of the comments on. If you absolutely must know a detail or two, feel free to look me up (I’m Glen, I do math, I work at the University of Wollongong) and get in touch. Ask me a pointed question and I’ll give a pointed answer, if I can.

  2. Lazy media is an axiom these days—no proof is required. However, this Pythagoras theorem proof is very nice.

  3. Why would “any proof of Pythagoras incorporating similarity … might be properly called a trigonometry proof”?

    1. I’m a bit skeptical of this as well. Similarity is a geometric concept, and although you could call it trigonometric, I don’t see how that’s much different from calling Pythagoras’ theorem in general trigonometric because it involves right angled triangles.

      In fact, I’m pretty sure that some of the propositions in Euclid’s Elements were about similarity, and Euclid’s Elements (although it is flawed [though much less than any VCE textbook] and incomplete by modern standards) is clearly a book about geometry, not about trigonometry.

    2. Thanks, Terry. It’s not my fight, and maybe read the linked cut-the-knot discussion, but at its heart trig is capturing the similarity of right-angled triangles. Then, the trig can be used to capture the similarity of all triangles. It might end up being a little contrived, but I don’t see it would be that hard to rephrase similarity arguments as trig arguments.

  4. It’s no fun being the party-pooper.

    Go back to Euclid. (Ouch! One might as well try defending James Cook, or Bob Hawke.)

    What many claim to be their “favourite” proof (“Drop the perpendicular from the right angle to the hypotenuse”) has the same “flaw” as the proposed “proof”.

    I don’t mean that it is “wrong”; but rather that it is misguided – or perhaps mathematically tasteless.

    A proof depends on assumptions.
    At the level of Pythagoras, this comes down to axioms.

    There is no agreed hierarchy of axioms.
    But nearly 200 years of experience with non-euclidean geometry suggests a natural hierarchy – which is only partly visible in Hilbert’s 1899 axiomatisation (“Die Grundlagen der Geometrie”).

    My version (for teachers) is in Chapter 5 of “The essence of mathematics through elementary problems” https://www.openbookpublishers.com/books/10.11647/obp.0168 .
    The aim there was to suppress those axioms that, while important, may appear gratuitous at high school level (“Equals added to equals are equal”, etc.).
    There are then three essential axioms: (i) that relating to congruence; (ii) that relating to parallels; (iii) similarity.
    The axiom of parallels (no matter how it is stated) is much more subtle than that of congruence.
    The axiom of similarity then brings in the full force of real numbers (in disguise).

    As Euclid’s proof of Pythagoras shows, Pythagoras Theorem requires (i) congruence and (ii) parallels; but *not* (iii) similarity. Hence any proof that appeals to similarity is using a hammer to crack a nut.

    1. Thanks, Tony. I take it, then, you’re not so thrilled with the calculus proof of Pythagoras?

      I’ve already confessed my heresy, that I’m not a fan of Euclidean geometry. So, I’m not so fussed about the hammer aspect, but I take your point.

      But I think there’s a more general and interesting question:

      How should one approach proof (or “proof”) in school mathematics?

      To which the answer “Occasionally, would be an improvement” springs to mind.

      But, ignoring the depressing reality, I think it’s an interesting and non-obvious question, at least to me. I’ve been vaguely pondering this recently, for a couple reasons. Maybe I’ll put up a separate post, so people can at least beat up on the heretic.

      1. I try to introduce my Year 9 students to proof. My approach is to give them the occasional example of a proof and explain why it is a proof. I do not formally define the meaning of “proof”; I just say “this is a proof”, “this is another proof” etc.

        Just this week, at the end of a term devoted to indices, I showed them Euclid’s proof that \sqrt{2} is irrational. I’d say that solving a Sudoku puzzle without guessing involves proof – students love this exercise.

        BTW, I came across an interesting example by Smullyan the other day.

        Given (1) Everyone loves my baby. (2) My baby loves only me. Prove (3) I am my own baby.

      1. The problem is ill-posed. The type of calendar being used must be specified.
        (I’m only being a little bit facetious – how to pose a problem is something students have to learn as part of problem solving and proof).

  5. “More generally, one has to ask what it means for a proof of Pythagoras to be a ‘trigonometry proof’, which is also discussed at cut-the-knot. For example, Grant has suggested that any proof of Pythagoras incorporating similarity, of which there are many, might be properly called a trigonometry proof. So, it is reasonable if not very generous to suggest that Johnson and Jackson were not the first to prove something that is not a thing.”

    Exactly. But here is a precedent for a proof “by trigonometry”, to the extent this is meaningful. There is one in Pogorelov’s grade 6 to 8 geometry textbook, which was widely used in the Soviet Union in the 1980s. The proof is preceded by a section defining the cosine of an acute angle. Here is the proof in full:

    Let ABC be the given right triangle, with right angle at C. Draw the altitude CD from the vertex C.

    By the definition of the cosine of an angle, \cos A = AD/AC = AC/AB, whence AB.AD = AC^2.

    Analogously, \cos B = BD/BC = BC/AB, whence AB.BD = BC^2.

    Adding the equations obtained term by term and noting that AD + DB = AB, we have AC^2 + BC^2 = AB(AD + DB) = AB^2, which proves the theorem.

    1. Thanks, J.D. I’ve attached a diagram.

      This is proof #6 in cut-the-knot’s list. They also discuss whether this is properly a “trigonometric proof”, which indirectly makes clear the silliness of the discussion.

      1. Right. I think this fundamentally has the makings of a pretty nice story. A pair of high school students have found a proof of Pythagoras which by most accounts is both entertaining and new.

        Unfortunately, the story is marred by the fact that the students seem to have taken Elisha Loomis’s bizarre pronouncement as being somehow reflective of mathematical opinion, and this has in turn been accepted uncritically by journalists with the effect (and I suspect the purpose) of knocking mathematicians down a peg in the eyes of lay readers.

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