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Refer to the second set of equations in the section about "Other Proofs - Proofs Using Similiar Triangles". The left hand term of each equation is incorrect/confusing. BC^2 is usually interpreted as "B times C-squared" but in this context what is wanted is "the square of the length of line segment BC". I realize that this section uses explicit multiplication symbols rather than multiplication-implied-by-adjacency, but it is still confusing in context of the rest of the page. I suggest the typography of the left-hand sides of these two equations be changed to [BC]^2 and [AC]^2 respectively, or use over-bars over all the line segment letter-pairs, which is the standard line-segment notation. Thanks, Craig H Collins 26 Jan 2023

 Fixed by adding a hatnote. D.Lazard (talk) 10:43, 26 January 2023 (UTC)[reply]

@Kencf0618 added this section, but the proof itself is not available in any of the links attached. 181.167.210.101 (talk) 16:24, 25 March 2023 (UTC)[reply]

I removed the section. D.Lazard (talk) 16:53, 25 March 2023 (UTC)[reply]

Granted, we'd all like to see the proof, but in the meantime, why shouldn't we at least note the two students who have made the claim?98.149.97.245 (talk) 14:29, 26 March 2023 (UTC)[reply]

To mention that, we need an independent source atesting that the claim is true and that the the proof is correct and not circular. As far as one can see for the provided summary, the use the law of sines, without establishing that this law can be proved without the Pythagorean theorem. This makes the claim very dubious. D.Lazard (talk) 16:30, 26 March 2023 (UTC)[reply]

It’s cool that these students got to present their work to mathematicians, but as of yet any claims about it are not verifiable, because no details have been published anywhere. Claims that this is the “first trigonometric proof” are clearly false, based on the students’ reliance on outdated sources / lack of a literature review. (Not trying to knock the students here.) Ultimately this seems mostly like a feel-good news report rather than significant mathematical news. If the news reports inspire other students to be curious and make their own discoveries (whether or not they turn out to be novel), that’s great. But Wikipedia shouldn’t exaggerate. –jacobolus (t) 20:00, 26 March 2023 (UTC)[reply]

Here's a reference to a non-circular proof using trigonometry (Zimba, 2009):

https://forumgeom.fau.edu/FG2009volume9/FG200925.pdf Tkircher (talk) 18:13, 26 March 2023 (UTC)[reply]

Upload the triangles to Wikimedia Commons. Youtuber reconstructed the proof. https://m.youtube.com/watch?v=nQD6lDwFmCc71.201.78.227 (talk) 05:28, 1 April 2023 (UTC)[reply]

Proof by Jackson and Johnson (2023): I support the inclusion of the trigonometric proofs in the main article.^[1][2] The proof by Zimba in 2009 is correct.^[2] Possibly others have also noticed the error in Loomis book,^[3] however, it is great that now the high school students have provided another quite beautiful proof.^[1] I have prepared an SVG sketch of their construction and already have uploaded it to Wikipedia commons:

The proof is valid for almost all right triangles, except for the case when a=b, i.e. Isosceles Right Triangle.

For the case when ${\displaystyle a\neq b}$, we can always choose to orient the right triangle ${\displaystyle \triangle ABC}$ so that a<b as shown in the figure. Then we reflect the right triangle to obtain point D. We extend leg BE perpendicular to AB, and extend AD until it crosses BE, here is why it is necessary for ${\displaystyle 2\alpha <{\frac {\pi }{2}}}$. Then, we need to sum a convergent infinite geometric series where ${\displaystyle {\frac {a^{2}}{b^{2}}}<1}$ in order to compute ${\displaystyle \sin \left(2\alpha \right)}$ from the large right triangle ${\displaystyle \triangle ABE}$:

${\displaystyle \sin \left(2\alpha \right)={\frac {\overline {BE}}{\overline {AE}}}={\frac {2c{\frac {a}{b}}\left(1+{\frac {a^{2}}{b^{2}}}+\left({\frac {a^{2}}{b^{2}}}\right)^{2}+\left({\frac {a^{2}}{b^{2}}}\right)^{3}+\ldots \right)}{c+2c{\frac {a^{2}}{b^{2}}}\left(1+{\frac {a^{2}}{b^{2}}}+\left({\frac {a^{2}}{b^{2}}}\right)^{2}+\left({\frac {a^{2}}{b^{2}}}\right)^{3}+\ldots \right)}}={\frac {2c{\frac {a}{b}}\left({\frac {1}{1-{\frac {a^{2}}{b^{2}}}}}\right)}{c+2c{\frac {a^{2}}{b^{2}}}\left({\frac {1}{1-{\frac {a^{2}}{b^{2}}}}}\right)}}={\frac {2ab}{a^{2}+b^{2}}}}$

Finally, we employ the Law of sines in ${\displaystyle \triangle ABD}$ to find out:

${\displaystyle {\frac {2a}{\sin \left(2\alpha \right)}}={\frac {c}{\sin \beta }}}$

which upon substituion with ${\displaystyle \sin \beta ={\frac {b}{c}}}$ and ${\displaystyle \sin \left(2\alpha \right)={\frac {\overline {BE}}{\overline {AE}}}}$ gives the Pythagorean theorem:

${\displaystyle a^{2}+b^{2}=c^{2}}$.

For the special case when ${\displaystyle a=b}$, the geometric series does not converge because ${\displaystyle {\frac {a^{2}}{b^{2}}}=1}$, however, the proof is purely algebraic using the areas of triangles ${\displaystyle \triangle ABC}$, ${\displaystyle \triangle ACD}$ and ${\displaystyle \triangle ABD}$, namely: ${\displaystyle {\frac {ab}{2}}+{\frac {ab}{2}}={\frac {c^{2}}{2}}}$, but since ${\displaystyle a=b}$, it follows that ${\displaystyle {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}}={\frac {c^{2}}{2}}}$.

Now, I would like to roast the text written by Loomis on page 244 in his book,^[3] freely accessible from ERIC:

NO TRIGONOMETRIC PROOFS

Facing forward the thoughtful reader may raise the question: Are there any proofs based upon the science of trigonometry or analytical geometry?

There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem; because of this theorem we say ${\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}$, etc. Trigonometry is because the Pythagorean Theorem is.

This shows poor understanding by Elisha Scott Loomis of what trigonometry is. The sine and cosine are originally usually defined (in high school textbooks) as ratio of two sides inside a right triangle and do not necessitate any knowledge of the equality ${\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}$. Apparently the mistake by author, Loomis, was clearly pointed out by Zimba in 2009^[2] but it is possible that others have been aware of this error even before 2009.

I hope that part of my notes can be reused by other editors to update the main article. Danko Georgiev (talk) 22:43, 1 April 2023 (UTC)[reply]

Your Jackson and Johnson link is just an announcement of a presentation with no details. Where’d the figure come from? –jacobolus (t) 02:28, 2 April 2023 (UTC)[reply]

sine and cosine are originally defined as ratio of two sides inside a right triangle – what do you mean by “originally defined”? Historically sine and cosine were defined as particular line segments relative to a reference circle. –jacobolus (t) 02:29, 2 April 2023 (UTC)[reply]

The figure comes from the news video from WWL-TV, exactly frame 1:30 min ^[4] Of course, the labeling and notation in my Figure is mine, but I have prepared it in the most pedagogical style that I came up with for possible usage in Wikipedia.

The "original" should have been: "the first definition ever encountered in high school textbook by an average person living on Earth". See, Wikibooks.org: High School Trigonometry/Defining Trigonometric Functions/The Sine, Cosine, and Tangent Functions. In other words, I was not talking about "historical" definitions, as historical events have no bearing on whether a modern mathematical proof is circular or not. By the way, what I have meant as "original definition" of sine is already in the Figure: ${\displaystyle \sin \beta ={\frac {b}{c}}}$ Danko Georgiev (talk) 08:43, 2 April 2023 (UTC)[reply]

A Wikipedian's speculative reconstruction of an unpublished presentation based on a blurry slide shown on a TV news program doesn’t really seem like it meets

WP:OR. Why don’t we just wait for these two students to publish their work? –jacobolus (t) 09:02, 2 April 2023 (UTC)[reply

]

"speculative reconstruction" - you are the most pleasant person on Earth, so I will tell you this: if you are incompetent to check whether a mathematical proof is correct or not, you should not edit mathematical pages. Also, my reconstruction is "exact" and there is nothing speculative in it. Danko Georgiev (talk) 09:14, 2 April 2023 (UTC)[reply]

I do not know whether the above proof is that of the two students (this is the meaning of “speculative”). But I see that, although rather ingenious, this proof uses two tools that do not belong to trigonometry, namely the concept of the sum of an infinite series, and the parallel postulate, which is equivalent with the widely used fact that the sum of the acute angles of a right triangle is ${\displaystyle \pi /2.}$ So this proof does not add nothing to the classical proof of the Pythagorean theorem, directly based on the parallel postulate, and nothing allows saying that this is a trigonometric proof. D.Lazard (talk) 10:06, 2 April 2023 (UTC)[reply]

This proof is "trigonometric" because it uses the definition of "sine" to compute the lengths of the sides in the infinite chain or right triangles. Also, pointing out that the proof depends on the parallel postulate is a dismissive and meaningless remark, and it is not different from making the remark that the proof also depends on the Peano axioms of arithmetic, because one uses addition, subtraction, multiplication and division. To that, I would just say: "so what?". In the axiomatization of any theory, one has the freedom to choose "what is an axiom" and "what is a theorem", i.e., there are some axiomatizations in which a certain statement is an axiom, and there are other axiomatizations in which the same statement in proven as a theorem from some other axioms. In summary, the proof by the New Orleans students is "trigonometric" because it heavily relies on repeated use of the definition of sine as ratio of two sides in right triangle. The rest in the proof is just algebra, e.g. Peano axioms which have nothing to do with the Pythagorean theorem. Also, the issue is not so much about the new proof, but the toxic culture by Wikipedia editors who are incompetent, but bully others as explained by Zimba in his article. And another issue, is the apparently influential error done by Elisha Scott Loomis which is mindlessly recited by others, for more see User:Danko_Georgiev/sandbox. Danko Georgiev (talk) 10:27, 2 April 2023 (UTC)[reply]

Every proof of the Pythagorean theorem necessarily relies on the parallel postulate (often filtered through intermediate concepts like a notion of similar triangles of different sizes). The two are logically equivalent and removing the parallel postulate breaks the Pythagorean relation. –jacobolus (t) 18:15, 2 April 2023 (UTC)[reply]

It is “speculative” in the sense that you are only working from one blurry photograph of one slide of a talk for which you are missing the other slides and the oral content of the talk. It is impossible to verify that the reconstruction is the same as the original. –jacobolus (t) 18:16, 2 April 2023 (UTC)[reply]

The reason I asked about "original definition" was not just to nitpick, but because the definitions and conceptual scope here matter a lot if we are trying to figure out which concepts are built on which others. If you read old geometry books or old trigonometry books (up through the 17th century or so) they do not have a concept of lengths as numbers or even ratios of lengths as numbers, but only a concept of pairs of ratios of straight lines (what Euclid calls a 'straight line' we now consider in terms of modern concepts to mean the length of a line segment) in proportion. So you can have AB : CD :: EF : GH, for straight lines AB &c., but it was not considered meaningful to write AB / CD or AB × CD (the former could be used as a ratio in proportion and the latter would be instead described as "the rectangle with sides AB and CD or the like). (I would link the relevant Wikipedia articles but they currently do a very poor job of explaining the conceptual distinctions / historical development).

Trigonometry was originally developed by Hipparchus and Ptolemy inter alia, and later by Indians, Arabs, & al. in the form of spherical trigonometry, a branch of astronomy, first with tables of chords and later with tables of sines describing (approximately) the relation between arc length and chord length. From what I understand, the concepts were not entirely well developed/established in the style of Euclid, and were treated as a kind of applied/practical approximate subject, not really part of geometry per se. In antiquity what we now do with "planar trigonometry" (of the style found in high school books) would instead be accomplished by Euclid-style constructions / theorems. For example Elements propositions 2.12–13 are what we now call the law of cosines:

2.12: In obtuse-angled triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.

2.13: In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle.

But notice there's no concept of angle measure or cosine (some scalar number as the ratio of sides) involved here.

So the main reason that there were historically no "trigonometric proofs" of the Pythagorean theorem is that the subject of trigonometry was treated as part of a separate tradition with separate methods, standards, and scope than the subject of geometry.

If you want to call a proof "trigonometric" that depends on first carefully defining what "trigonometric" means, then establishing all of the relevant theorems of such trigonometry independent from the Pythagorean theorem. This is going to necessarily be a bit of an affectation, because "trigonometry" in the sense of ratios in triangles is based on the concept of similar right-angled triangles which is logically equivalent to the Pythagorean theorem. So it’s always going to be an indirect route to proving something that you could just as easily prove more directly from the same axioms. But of course, the same can also be said for many other kinds of proofs of the Pythagorean theorem, such as those built on integral calculus.

Anyway, I am not just trying to be a party pooper. I will wholeheartedly agree that it’s great when high school students get excited about math. –jacobolus (t) 21:14, 2 April 2023 (UTC)[reply]

"because "trigonometry" in the sense of ratios in triangles is based on the concept of similar right-angled triangles which is logically equivalent to the Pythagorean theorem" -- this is false. To define ratio of two sides you do NOT need the the Pythagorean theorem. And I can prove it to you immediately: to define the ratio, you need only the notion of "scale" and "similarity". For example, I use a triangle that does not have a right angle and define the notions of "scaling" and "similarity". These cannot be "presuming" right angle, because you will arrive at contradiction when talking about similar non-right triangles. This establishes, that the concepts "scale" and "similarity" are independent of the right triangle. Now, it is easy to define the functions "sine" and "cosine" as ratios, but you specify that you will be using right triangle. These functions are independent from the Pythagorean theorem because the "scale" and "similarity" were independent and the only new thing added was the "right angle". Using a simple argument of removal of independent things, if you say that the definition of "sine" and "cosine" already imply the Pythagorean theorem, then you will be stating in effect that the existence of the right triangle (or right angle) is implying the Pythagorean theorem, which is provably false -- namely, there are non-Euclidean geometries in which the Pythagorean theorem is false, but there exist right triangles and right angles. In summary, I have just proved to you that you are talking nonsense. The definition of sine and cosine as is usually done in high school textbooks, does not imply the Pythagorean theorem. Even, I can give you counterexample so that it becomes crystal clear to you: draw a triangle on a sphere! The definition of "sine" and "cosine" in the curved triangle is perfectly well defined as the ratio of the curved triangle sides. So, the Pythagorean theorem is absolutely false on the sphere and the "sine" and "cosine" functions have nothing to do with the "sine" and cosine" of the Euclidean space. So back-off: the definition of the sine and cosine as ratio of sides depends on the curvature of the space. I am using the parallel postulate to impose "flatness", not the Pythagorean theorem. What you are claiming is demonstrably false, and has nothing to say on the correctness of the proof that I have attributed to Jackson and Johnson above. Danko Georgiev (talk) 00:06, 3 April 2023 (UTC)[reply]

P.S. in case if you are wondering what I am saying is this: "the ratio definitions of sine and cosine" are independent on the Pythagorean theorem because in curved space sin^2 + cos^2 is NOT 1. So I feel that we have identified the origin of the whole debate, namely, there are different non-equivalent definitions of the trigonometric functions. If you define "sin^2 := 1 - cos^2", then this is indeed the same as the Pythagorean theorem. But if you define the sine function as ratio, this is much more general definition that is curvature-dependent and is NOT equivalent to the Pythagorean theorem. Do you agree? Danko Georgiev (talk) 00:23, 3 April 2023 (UTC)[reply]

You are misunderstanding/misstating what I said. Perhaps I was not clear enough. To define ratio of two sides you do NOT need the the Pythagorean theorem – nobody has said this. However, to establish that the ratio of the sides is uniquely determined by the angle, and vice versa, you need the parallel postulate or something logically equivalent (the Pythagorean relation is one possible axiom you could start with; or you could start with some statement about similar triangles as your axiom; or various others). ... define the notions of "scaling" and "similarity" you are saying the same thing I just said: you could establish definitions of sine etc. by first establishing a notion of similar triangles.

there are non-Euclidean geometries in which the Pythagorean theorem is false, but there exist right triangles and right angles. [...] definition of the sine and cosine as ratio of sides depends on the curvature of the space You are repeating what I just said.

To be precise, in the context of the sphere, a right triangle with sides ${\displaystyle a,b,c}$ and right angle at ${\displaystyle C}$ where ${\displaystyle a=\tan {\tfrac {1}{2}}\angle BOC}$ etc. (${\displaystyle O}$ being the center of the sphere, and ${\displaystyle \angle BOC}$ meaning the measure of the central angle), the analogous Pythagorean theorem is instead the relation:

${\displaystyle a^{2}\boxplus b^{2}=c^{2}}$

where ${\displaystyle x\boxplus y:=(x+y)/(1+xy)=\tanh(\operatorname {artanh} x+\operatorname {artanh} y).}$ Because ${\displaystyle \boxplus }$ does not scale proportionally with both of its arguments (the way + does), there is no notion of similarity on the sphere. (The same formula holds in the hyperbolic plane, if you take ${\displaystyle a,b,c}$ to be the hyperbolic half-tangents instead of circular half-tangents of the sides. The Euclidean version is the limit of the spherical/hyperbolic formula when ${\displaystyle a^{2}b^{2}\ll 1.}$)

If we don't want to rely on a concept of angle measure, we can let ${\displaystyle A,B,C}$ be points on a sphere in Euclidean space with ${\displaystyle B^{\oslash }}$ meaning the point antipodes to ${\displaystyle B}$, and define half-tangents of sides like ${\displaystyle a=|(C-B)/(C-B^{\oslash })|}$ etc., where ${\displaystyle C-B}$ is a vector and the ${\displaystyle /}$ means the geometric quotient so ${\displaystyle a}$ is the magnitude of a unitless bivector oriented in the plane of ${\displaystyle B}$ and ${\displaystyle C.}$ We can even take the absolute value signs off if we want, because these bivectors square to scalars. But that's probably enough about the side question of Pythagoras in non-Euclidean planes. –jacobolus (t) 00:34, 3 April 2023 (UTC)[reply]

OK, thanks for the extra clarification. Euclid's parallel postulate is needed in the proof because you need to establish that the sides AE and BE intersect when ${\displaystyle 2\alpha <{\frac {\pi }{2}}}$. You need the "flatness" of the space to prove the Pythagorean theorem. There was no dispute that you need the parallel postulate at all. Also, risking to repeat myself, I do not object to take Pythagorean axiom and prove the parallel postulate as theorem, i.e., exchanging which of the two statements is axiom and which is theorem. What I do not understand is to what exactly you are objecting in the proof in order NOT to classify it as "trigonometric"? A side note, based on your mentioning of integral calculus and Bogomolny's page: to sum convergent infinite series you only need to define limits - in contrast, to define differential and integral calculus you need to add so much extra theory that I am not sure (1) that all of this extra stuff is necessary for the proof of the Pythagorean theorem, and (2) whether some of this extra stuff does not actually require the Pythagorean theorem. Frankly, I no longer understand what your concerns were, and what definitions you wanted to have from me. Just take a basic high school textbook of trigonometry and let us call it a night. Danko Georgiev (talk) 00:59, 3 April 2023 (UTC)[reply]

What I am saying is nobody has bothered too much in the past trying to invent "trigonometric" proofs here because there’s no particular reason (beyond amusement) to try to establish the tools of plane trigonometry independent from basic relations of Euclidean space (like the Pythagorean relation). Any trigonometric proof you end up with is more or less a slightly more cumbersome variant of an analogous proof you could make just in terms of similar triangles, and establishing the tools of trigonometry necessarily leans on already developing a concept of similar triangles. So whatever trigonometric proof you come up with somewhat has the flavor of a Rube Goldberg machine, adding extra indirection just for the sake of it. Whether such a proof is properly "trigonometric" or not is not really an empirical question but a semantic one, dependent on what someone means by "trigonometric proof", and whether the reader considers a proof that is same except for having some side ratios replaced with sines/cosines to be novel. –jacobolus (t) 01:12, 3 April 2023 (UTC)[reply]

Edited text with references on the "Trigonometric Proof of the Pythagorean Theorem by Jackson and Johnson (2023)" and "Nonexistence of trigonometric proofs of the Pythagorean Theorem claimed by Elisha Scott Loomis" can be copy-pasted from my sandbox: User:Danko_Georgiev/sandbox. Danko Georgiev (talk) 10:56, 2 April 2023 (UTC)[reply]

The content in your sandbox is about 5–10 times too long to be in scope for this article, in my opinion, in accord with

WP:DUE. If you cut it down to ~1–3 short paragraphs including the shortest exemplary trigonometric proof that can be found in published literature, with alternatives discussed (not spelled out) in footnotes, it could fit in a section "Trigonometric proofs". This article currently picks about 3 or 4 exemplary proofs (the oldest and most famous ones) and then compresses discussion of the remaining hundreds of alternative proofs into a paragraph or two and maybe a couple figures about each other broad category. That seems like about the right strategy to keep the article legible to an anticipated typical reader, and keep the narrative at least somewhat moving along. –jacobolus (t) 19:05, 2 April 2023 (UTC)[reply

]

Correcting a misleading statement by Loomis or removing it from the article is one thing, including the new proof is another. As long as the proof is not published in journal or book, I don't really see a good reason to include it here as an inclusion at this point collides with various Wikipedia policies. This has nothing to do with correctness of the proof or whether one considers it trigonometric or not. And even if such a publication has become available, that is still no reason for an automatic inclusion. Keep in mind our article contains only a small subset of the available proofs of the theorem, which primarily means those most common in literature and/or might representative for a larger class of proofs.--Kmhkmh (talk) 10:58, 2 April 2023 (UTC)[reply]

The inclusion of the proof by Jackson and Johnson in the main article will serve several purposes: (1) it is mathematically "beautiful" construction, which cannot be said to the majority of say 370 proof in the book by Loomis. For example, if you just open the free online PDF copy of the Loomis book and browse through it, you may get the impression that some of the proofs are over-crowded constructions whose only purpose is to increase the count of proofs; (2) the fact that the proof is produced by

high school students is remarkable in itself in terms of authorship, and (3) it is important to point out that there is a toxic environment in mathematics, so much so that since 1927 when the book by Loomis was published, it has been used to suppress correct arguments by simply "quoting" page 244, which says in all caps that there are "NO TRIGONOMETRIC PROOFS". With regard to notability concerns, the fact that the proof by Jackson and Johnson has been featured in The Guardian^[5] is sufficient to merit coverage in Wikipedia. If you count the number of years since 1927, you can determine that the event of high school students disproving claims published in academic mathematical books is one event per 96 years. Danko Georgiev (talk) 11:28, 2 April 2023 (UTC)[reply

]

As far as I understand this article didn’t ever repeat Loomis’s claim, and whether a proof of the Pythagorean theorem can be "trigonometric" or not frankly doesn't seem that important (to me personally), being to a substantial degree a semantic dispute based on the imprecise definition of the word "trigonometric". But while we are here, plenty of sources can be found about "trigonometric" proofs and disputing Loomis's claim. Several are discussed/linked from Bogomolny (2012) "More Trigonometric Proofs of the Pythagorean Theorem". If someone wants to add a section about Trigonometric proofs after the section about

Pythagorean Theorem § Algebraic proofs I would be indifferent to it. But it should stick to published claims, not original research by Wikipedians. Encouraging high school students is a valuable goal, but trying to force discussion of high school students into Wikipedia articles is not really the best mechanism for that in my opinion. –jacobolus (t) 18:43, 2 April 2023 (UTC)[reply

]

The proof given by Bogomolny is circular.

1. Consider a point ${\displaystyle (1,\theta )}$.
2. ${\displaystyle \cos \theta }$ and ${\displaystyle \sin \theta }$ are the lengths of the legs of a right triangle.
3. Their projections onto the hypotenuse have lengths ${\displaystyle \cos ^{2}\theta }$ and ${\displaystyle \sin ^{2}\theta }$.
4. Therefore, ${\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}$

To compute distances from coordinates ${\displaystyle (1,\theta )}$ e.g. to the origin ${\displaystyle (0,0)}$ (or any other point) requires the use of the distance formula ${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}$ which is exactly the Pythagorean theorem. There is no way to proceed from coordinates to lengths, because one does not have a distance function (or metric) that follows from these coordinates. Imposing the Euclidean distance formula ${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}$ is to assume the Pythagorean theorem from the start, hence, it is a perfect example of circular reasoning.

By the way, the "nice" thing about the trigonometric proof by Jackson and Johnson is that you just need a ruler and compass to construct the right triangles, and then just some straightforward arithmetic. You may be summing infinite series, but at least you are not doing the "circular" business. Next time when you provide references, at least be sure that you are not putting forward questionable materials. Danko Georgiev (talk) 21:17, 2 April 2023 (UTC)[reply]

You put the distance function in there, not Bogomolny. But I will agree that this page does not go through all of the details necessary to demonstrate to a skeptical reader that the proof is not circular. That would take developing many more pages of preliminary results. In particular, if you want to talk about an angle measure ${\displaystyle \theta }$ as an arclength of a unit-radius circle, and put it on solid rigorous footing, you probably need to bring in integral calculus. –jacobolus (t) 21:19, 2 April 2023 (UTC)[reply]

I did not put anything anywhere. I ask only the question: how do you go from "coordinates" to "lengths". Option 1: you tell me that you use the distance function that I wrote above and you shoot yourself in the foot. Option 2: you tell me that I do not have a formula but I need to use some other method using ruler and compass. Then I will reply this: "who are you trying to bamboozle with the "coordinate terminology"?? If you put coordinates in point 1 in the proof, and then you are telling me that I cannot use these coordinates for anything, then you are not providing any proof. In fact, by inserting terminology that I cannot use later for anything I am going to conclude that you have no proof at all. period. Danko Georgiev (talk) 21:32, 2 April 2023 (UTC)[reply]

Note this in polar coordinates, not rectangular ones. I think the author (in the context of a message to a newsgroup) was compressing their description of a point on the circle for the sake of concision/clarity because a detailed rigorous elaboration would be cumbersome. The ruler-and-compass way would be to just pick out a point on the circle. But whether this avoids the Pythagorean theorem or not is going to depend on making careful definitions of sine and cosine when starting from a point on the circle. –jacobolus (t) 21:35, 2 April 2023 (UTC)[reply]

Wow, for the sake of clarity the argument has become so transparent that I can no longer see it. Wonderful job! Danko Georgiev (talk) 21:41, 2 April 2023 (UTC)[reply]

P.S. can we try to avoid the sarcasm, insults, etc. here? –jacobolus (t) 21:47, 2 April 2023 (UTC)[reply]

"That would take developing many more pages of preliminary results" -- this is exactly the point at which I stop talking with somebody. Danko Georgiev (talk) 21:50, 2 April 2023 (UTC)[reply]

Your proof above also requires many unstated preliminary results. You need to carefully (1) define what you mean by a side labeled by some algebraic expression and prove that it is meaningful to perform algebra on your labels so defined, (2) define what you mean by an angle of label e.g. ${\displaystyle \alpha ,}$ (3) define what you mean by ${\displaystyle \sin \alpha ,}$ and prove that it is equal to the quotient of two algebraically expressed sides irrespective of the particular sides/angles involved, (4) define what it means to add angles, (5) define what you mean by ${\displaystyle \pi /2,}$ (6) establish that it is meaningful to sum an infinite series of algebraic expressions and have the sum be a meaningful label for the side that your segments placed end-to-end converge to, (7) prove that your specific series are convergent, (8) prove the law of sines, etc. If we think hard we could probably come up with at least half a dozen more prerequisite steps. If you rely only on what is found in a typical high school textbook, the result will assuredly have some holes that are not rigorously established to the satisfaction of modern mathematicians. Making sure that none of these involve any invocation of the Pythagorean theorem is possible but you’ll have to be careful about it because books describing all of these concepts usually take the Pythagorean theorem for granted and don’t bother avoiding all of its consequences or re-deriving them in a loopy way. Indeed, modern rigorous geometry books often side-step this whole mess by starting with a coordinate plane with "Euclidean structure", i.e. some distance function, bilinear form, or quadratic form, so that the Pythagorean theorem is essentially taken as an axiom, or for a general right triangle involves some trivial arithmetic. –jacobolus (t) 22:10, 2 April 2023 (UTC)[reply]

Note, I’m not saying you should do this. My point is just that to prove a theorem we need to establish some context of previously proven results considered fair game to built from. Usually in proving new theorems mathematicians are happy to use any already-accepted results from anywhere in mathematics, while in writing a textbook an author will pick some set of assumed prerequisites and then try to make a logically rigorous path from those to everything else in the book, without internal circularity or dependence on concepts that are out of scope. But when making new decontextualized proofs of old theorems, especially with a constraint like “can’t rely on anything involving the Pythagorean theorem”, some kind of context needs to be established (otherwise readers such as yourself "can no longer see" the argument). The concepts and methods you are using here (or that the authors in Bogomolny's message group used) are not defined in the original context of the Pythagorean theorem, such as Euclid's Elements, which limits itself to a few simple axioms and only "geometric" reasoning, with no concept of angle measure and any algebra dressed up as geometry. –jacobolus (t) 22:55, 2 April 2023 (UTC)[reply]

But there is no need to refer Euclid as the contextual framework (or prequisite) for our article and no need to rely on other historical contexts/prequisites for the proofs provided in our article. Instead we consider as prequisites/context what is currently commonly taught in schools (the knowledge most of our readers start off with) and what is given in modern geometry books (rather than Euclid). Euclid and other older sources (and their notations and approaches) are primarily of interest for historical reasons/aspects, but not really in terms of giving accessible proofs.--Kmhkmh (talk) 09:43, 3 April 2023 (UTC)[reply]

The reason to bring up Euclid is that the whole of Book 1 of Elements leads up to the proof of the Pythagorean Theorem. (Indeed I think we could do a significantly better job of explaining the history/context about the proof of proposition I.47 in this article; I should perhaps try to write some more about it.) You can think of this as a proof from scratch. So if you look at Euclid you can see precisely what is required from axioms right up through the final theorem. On the other hand, trigonometry books, calculus books, etc. generally take the Pythagorean theorem and various other theorems built on top for granted already. –jacobolus (t) 14:54, 3 April 2023 (UTC)[reply]

I understand that. However I'm just saying that at least from a non-historical perspective this isn't really the appropriate approach, instead one should use modern (synthetic) geometry books starting from scratch (essentially Hilbert and later) or as far as our WP article is concerned use as prequisite/context what is taught in middle/high school math. This matters for the discussion above as you do not have to deal with concepts in the same order as Euclid or some other historic sources did, in particular you do not have to consider historic approaches to trigonometry, but can you can approach it via the ratios in similar right angled triangles (as given in the forum geometricorum article above). Or in the bigger picture starting from scratch you can go the following route (which in my experience school geometry usually does, without necessarily spelling it out explicity): axioms (essentially still the same as Euclid's) -> concept of areas -> area of a triangles -> intercept theorem/thales' (basic similarity) theorem -> similar triangles -> Pythagorean theorem. Now instead of using similar triangles directly to prove the Pythagorean theorem (this proof is in our article), one can introduce sine and cosine first based on similar triangles and then use trigonometry to prove the Pythagorean theorem. That yields you a path to the Pythagorean from scratch (or first principles) but in different order than Euclid. And as I said from today's perspective you still want from scratch but there is no need to follow Euclid's order. Moreover with regards to the order (or prequisites) we should follow popular modern geometry books rather then Euclid. The order I outlined above btw was taken from Hans Schupp's Elementargeometrie (UTB, 1977).--Kmhkmh (talk) 23:10, 3 April 2023 (UTC)[reply]

Here you are: (1) Ingredients of the proof (2) Steps of the actual proof. Nobody is required to read Euclid's Elements in order to understand what the ingredients of the proof are and what are the steps in which they are used. Danko Georgiev (talk) 09:54, 3 April 2023 (UTC)[reply]

@Danko Georgiev: Imho neither of the three arguments for inclusion you raised above do really apply:

• a) Whether a proof is in Loomis or not (or the quality of Loomis' collection) was never an argument for including a proof here. In fact I'd argue Loomis is completely irrelevant for the inclusion of a proof. A reason for inclusion instead is that a proof (which may or may not be included in Loomis' book) is that the proof is popular/appreciated in external publications (rather than primarily appreciated by some Wikipedians with (almost) no external publications).
• b) It is great that some high school students came up with a new proof, but that doesn't necessarily yield a reason to iclude the proof here. If that proof becomes popular in publications down the road, then we will have a strong reason to include it, but not before. Also we don't do promotional stuff, in a sense that we do not promote new topics/content in WP, but merely reflect/summarize which new content/topic gets promoted in external publications.
• c) Whether there is a toxic atmosphere in WP or not clearly has no bearing on the decision whether some content should be included or not. We include material we regard as encyclopedically relevant and not to avoid a potential impression of toxicity. While I agree that the atmosphere and content disputes in WP can be quite toxic and that rules and formalities are often (mindlessly) overemphasized, I don't believe that to be the case here. Moreover imho the fact encyclopedic writing by its very nature often can be a dry, sober and even boring affair is sometimes confused with toxicity. This is simply how writing in WP differs from writing books, journal articles, blogs, other Wikis, etc. Those places allow for enthusiastically writing about new material, promoting it, taking a personal viewpoint/opinion/focus, but WP (for the most part) does not.

--Kmhkmh (talk) 01:10, 3 April 2023 (UTC)[reply]

There are already several videos in YouTube on Jackson and Johnson. Users @User:Kmhkmh and @user:jacobolus objected to inclusion because "the high school students have not published their result yet." OK, fine! Do not include Jackson and Johnson inside the main article as long as you wish! However, it is important to fix the content of the main article, because there is a repeated construction in Proof using similar triangles and Einstein's proof by dissection without rearrangement. In fact, the very same construction can be used to give a trigonometric proof of the Pythagorean_theorem without the need of summation of infinite geometric series.

${\displaystyle \sin \alpha ={\frac {a}{c}}=\cos \beta }$

${\displaystyle \sin \beta ={\frac {b}{c}}=\cos \alpha }$

${\displaystyle c=b\cos \alpha +a\sin \alpha ={\frac {b^{2}}{c}}+{\frac {a^{2}}{c}}}$

${\displaystyle c^{2}=a^{2}+b^{2}\qquad ~~~\square }$

Already students in math.stackexchange ask whether this is a valid "trigonometric" proof and whether it is correct. There were a number of trolls who immediately claimed that the proof is "circular" which it is not. So, I highly recommend that this trigonometric proof is discussed alongside the Einstein's proof and the proof with similar triangles. All these 3 proofs are essentially 3 different viewpoints on the same construction. Also, my previous discussion with @user:jacobolus applies - it needs to be pointed out that the high school definition of "sine" and "cosine" as ratios require as a special intermediate step the proof that non-congruent similar triangles exist based on Euclid's parallel postulate. Thus, the trigonometric proof is not circular as the concept of "similarity" is already needed in the proof by similar triangles and also Einstein's derivation based on proportionality of areas. Danko Georgiev (talk) 15:31, 7 April 2023 (UTC)[reply]

Also, I would like to point out that the verbose textual drivel on Einstein's proof is not a proof at all, and will not be understandable to a high school student!!! Are you guys writing Wikipedia for yourselves? Bear in mind that that Wikipedia should not be understandable only for retired mathematicians, but to teenagers in high school too. So, if you do not object, I would like to insert the Trigonometric proof either inside Einstein's proof section or as a separate section immediately Einstein's proof section. Also, I would like to move all 3 proofs immediately following one after another. It makes no sense Einstein's proof not to be immediately following the proof with similar triangles. Also, being "Editor" implies that you should be taking care of the overall structure and logical flow of sections. I frankly do not understand who voted this article to be a "featured article" as the flow of logic is interrupted and related material appears all over the place. The identity of the two constructions mention by me is just one example of "duplicate" figure. In fact, from a single figure prepared by me above, one can write all 3 proofs in logical sequence one after another. Danko Georgiev (talk) 15:43, 7 April 2023 (UTC)[reply]

This is the same proof about which you said above: “The proof given by Bogomolny is circular.” (Except that version used a unit-length hypotenuse.) I agree with your updated take that it’s fine (and also already essentially included in the article, just without labeling the sides as sines/cosines per se). As we were discussing above, whether this counts as “trigonometric” and whether proofs by similar triangles which re-label the sides as sines and cosines are different from the same proofs just using labels a, b, c is a semantic question rather than a mathematical question.

... fix the content of the main article, – as far as I can tell the main article doesn’t make any false claims about this and doesn’t need to be “fixed”. But as I said above, a short section about “trigonometric proofs” could probably be added. –jacobolus (t) 15:49, 7 April 2023 (UTC)[reply]

The proof given by Bogomolny is circular because in step (1) he introduces "coordinates", in step (2) he uses "lengths", and in steps (3) and (4) he claims to prove the Pythagorean theorem. The circular reasoning comes from the fact that one cannot use the "coordinates" for calculation of "lengths" without using the Pythagorean theorem. If you already have the Pythagorean theorem in use between steps (1) and (2), you no longer need to prove it again in steps (3) or (4). The length of the line ${\displaystyle c}$ connecting two points with cartesian coordinates ${\displaystyle (a,0)}$ and ${\displaystyle (0,b)}$ is given by the Pythagorean theorem ${\displaystyle c={\sqrt {\left(a-0\right)^{2}+\left(0-b\right)^{2}}}}$. The length of the line ${\displaystyle c}$ connecting two points with polar coordinates ${\displaystyle (a,0)}$ and ${\displaystyle (b,{\frac {\pi }{2}})}$ is given by the Pythagorean theorem ${\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \left({\frac {\pi }{2}}\right)}}}$. Are we done with Bogomolny's proof? It is not the same as the proof given above by me because I do not introduce "coordinates" as step (1) in my proof. Also, in Eastern Europe where Bogomolny and myself are born and educated, if you introduce irrelevant stuff that you do not make use of in a math exam, you will get minus points from your overall score. In mathematical proofs you are not supposed to define and introduce stuff that you do not use or cannot use in your proof. Danko Georgiev (talk) 16:19, 7 April 2023 (UTC)[reply]

“Coordinates" is a red herring. You are misinterpreting the statement, which is a shorthand for a geometrical argument essentially identical to the one you listed here. ––jacobolus (t) 19:43, 8 April 2023 (UTC)[reply]

I have now moved the three proofs in sequential order, one after another, and have replaced a somewhat duplicate image using the more informative one with labeled height to hypotenuse and lengths of sides. I have wikified a bit, but if the text can be made more comprehensible to a high school student, please go ahead and improve it. Danko Georgiev (talk) 18:06, 7 April 2023 (UTC)[reply]

BBC News -

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.

https://www.theguardian.com/us-news/2023/apr/07/new-orleans-teens-pythagorean-theory

Depending on how this sorts out we might want to add something about it to the article.

- 186.215.16.37 (talk) 18:34, 8 April 2023 (UTC)[reply]

This is lengty discussed above. D.Lazard (talk) 18:54, 8 April 2023 (UTC)[reply]

Actually, Euclid has two proofs of the Pythagorean Theorem, namely Book I, Proposition 47 and Book VI, Proposition 31. The so-called Einstein's proof is, in fact, Euclid's second proof. Furthermore, this proof also appears in Loomis's collection of 1968, being attributed to Stanley Jashemski (contradicting Schroeder). Should this proof be credited to Einstein? Isn't this an example of Stigler's law of eponymy? Is Schroeder's reference more reliable than Loomis's? I personally think it should be entitled "Euclid's Second Proof" and perhaps state later that it was rediscovered by Stanley Jashemski (quoting Loomis) and/or Albert Einstein (quoting Schroeder). — Preceding unsigned comment added by 109.253.191.18 (talk) 20:36, 8 April 2023 (UTC)[reply]

This article mentions the similarity-based proof from VI.31 but only in the "generalizations" section. It’s not quite the same as "Einstein's" proof (credited to Einstein by Schroeder (1991) because Einstein reportedly found it independently at age 11, not necessarily because he was the first to discover it) which is based on a dissection of the original right triangle's area, not just any arbitrary similar figures erected on each side. But I agree with you the VI.31 proof should probably be discussed earlier. There should probably also be some discussion about why Euclid included the first proof using only the propositions from Book I, considering that the second proof is so much less tricky to follow once the Book V theory of proportions has been developed. –jacobolus (t) 21:10, 8 April 2023 (UTC)[reply]

I did not realise VI.31 is mentioned in the article, thanks for the correction. However, I still think that the proof of VI.31 is Einstein's proof word-for-word [1] and that it is unfair to attribute it to Einstein. Furthermore, this attribution spreads the misconception that confronts Einstein's "simple" proof against Euclid's "overcomplicated" proof, which is very unfortunate knowing about VI.31. 109.253.192.93 (talk) 21:56, 9 April 2023 (UTC)[reply]

Continuing from my previous inconclusive reply (I am sorry my previous reply was so short and unconstructive), yes, my main suggested correction to the article is that VI.31 should be mentioned in the proofs section (and not only as a generalisation) because Euclid provided an alternative proof in VI.31 without citing I.47. Also, I agree that it might be nice to add a discussion briefly explaining why Euclid gave two different proofs. However, before making these modifications, it is worth clarifying whether or not Einstein's proof is the same as Euclid's second proof.

If they are the same, as I think, they should share the same item. In that case, I suggest that this proof should be retitled as "Euclid's second proof", indicating that it is also known as Einstein's proof because it was rediscovered by Einstein. Alternatively, it can keep its current title, but the article should make clear that it is already VI.31 and it is not original of Einstein — so the reader does not get the misconception I mention in my previous comment.

Otherwise, if I am wrong and Einstein's proof and VI.31 are different, then we should add something about VI.31 in the proofs section: either along with I.47 or as a new item. 109.253.192.93 (talk) 17:19, 10 April 2023 (UTC)[reply]

They are not really the same. Both use the concept of similarity, but "Einstein's" proof (whoever may have first written it) is about decomposition of the triangle (that is, making one area by physically pasting the other two areas together), whereas Euclid's proof in VI.31 is based on using a sum of lengths of the two parts of side c, along with VI.19: "if three straight lines are proportional, then the first is to the third as the figure described on the first is to that which is similar and similarly described on the second." (In algebraic notation we might write this as ${\displaystyle a/b=b/c\implies a/c=ka^{2}/kb^{2},}$ though that is somewhat conceptually anachronistic.) –jacobolus (t) 18:13, 10 April 2023 (UTC)[reply]

Ok, you are completely right. Although they are very similar, they are not the same. I can see the difference now. In Einstein's proof one compares the squares with the triangles, while in Euclid's proof one compares the squares with the two parts of the hypotenuse (the projections of the legs). I am writing them below in a parallel way to remark the similarities and where the difference is. Thank you very much for your reply.

Einstein's proof: Let ABC be a right triangle with hypotenuse AC and BD be perpendicular to AC (with D point in AC). Note that ADB is the right triangle similar to ABC with hypotenuse AB and BDC is the right triangle similar to ABC with hypotenuse BC (by VI.8). Now, the square of AC is to the square of AB as ABC is to ADB and, similarly, the square of AC is to the square of BC as ABC is to BDC (by VI.20). Hence, ABC is to ADB,BDC as the square of AC is to the sum of the squares of AB and BC (by V.24). Now, the sum of ADB,BDC equals ABC, so the square of AC equals the sum of the squares of AB and BC.

Euclid's second proof: Let ABC be a right triangle with hypotenuse AC and BD be perpendicular to AC (with D point in AC). Note that ADB is the right triangle similar to ABC with hypotenuse AB and CDB is the right triangle similar to ABC with hypotenuse CB (by VI.8). Hence, AC is to AD as twice AC is to AB and, similarly, AC is to DC as twice AC is to BC. Thus, the square of AC is to the square of AB as AC is to AD and, similarly, the square of AC is to the square of BC as AC is to DC (by VI.20). Hence, AC is to AD,DC as the square of AC is to the sum of the squares of AB and CB (by V.24). Now, the sum of AD,DC equals AC, so the square of AC equals the sum of the squares of AB and CB. 109.253.210.184 (talk) 21:28, 10 April 2023 (UTC)[reply]

In my opinion this article would be improved by a bit of reorganization. I think it should start (in a section immediately after the lead) with a few diagram-heavy proof sketches and plain-language discussion about the basic historical and mathemtical context (and cut the "Other forms of the theorem" section which is trivial and distracting), explaining that we don't precisely know what the first proof may have been like, but mentioning some of the speculation about why Euclid included both the first proof I.47 as well as the second proof VI.31. I'm not sure what such an introductory section should be called; or maybe it could just be the top part of the proofs section.

Then all of the proofs should IMO be moved into a section "proofs", starting first with spelled-out proofs I.47 (including some discussion of the other propositions in book I which it is built on), VI.31 (explaining the propositions / methods on which it is based, and perhaps also describing the "Einstein" variant), then continuing to a few graphical re-arrangement proofs, and then several more subsections about others of different styles. ––jacobolus (t) 22:19, 10 April 2023 (UTC)[reply]

I completely agree, the article needs some reorganisation. Unfortunately, it is semiprotected and I cannot make editions. The section about "Other forms of the theorem" adds nothing, as you say. I would start with the "History" section first. Then, as you say, all the proofs should be given in one section called "Proofs", with an introduction indicating that this theorem has many proofs (it can cite Loomis for example) and that here we only collect a short amount. This section should start with a subsection called "Visual proofs" with a few diagram-heavy and animeted proofs — for instance, the proofs by rearrangement, dissection and area preserving shearing should be put together here. The next subsection should be "Euclid's proofs" containing I.31 and VI.31 and the speculation about why Euclid gave two proofs. After that, it can give as variations of VI.31 "Einstein's proof", "Proof using similar triangles" (which is a modern way of doing VI.31) and "Trigronometric proof using Einstein's construction" (which is just a rewriting of the previous one by replacing a/c by the sine and b/c by the cosine). Finally, it should end with a subsection about the algebraic proofs and a subsection about the analytic proofs. Next, the section "Consequences and uses of the theorem" should be probably retitled "Related results" and the "Converse" section could be a subsection of this section. The final section should be "Generalizations" and perhaps it should mention Parseval's identity at the end of "Inner product spaces" subsection. 109.253.210.184 (talk) 18:39, 11 April 2023 (UTC)[reply]

Why short Zimba trigonometric proof (main idea) from this revision https://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&oldid=1149322678#Jason_Zimba_trigonometric_proof%5B25%5D was deleted? @David Eppstein Kamil Kielczewski (talk) 19:14, 11 April 2023 (UTC)[reply]

There are literally hundreds of proofs of the theorem, maybe thousands. Picking out and including only one of these, sourced only to its primary publication, makes no sense, because there is no clear selection criterion for it that would not also cause us to also include hundreds of other proofs. We should only include proofs with significant historical recognition, not recent flash-in-the-pan media hype and even more not primary sourced but otherwise non-notable proofs vaguely connected to recent flash-in-the-pan media hype. —David Eppstein (talk) 19:39, 11 April 2023 (UTC)[reply]

I agree with David Eppstein; the Zimba proof is insufficiently noteworthy. —Quantling (talk | contribs) 19:49, 11 April 2023 (UTC)[reply]

The criterion for it is that is very short, simple and use only calculations without involving geometry (in direct way) like other proofs. So it can be very useful especially for people who hat not goot geometrical intuition (so we are dealing here with usability for a wider audience)

In the other side, for historical point of view, this is also first known trigonometrical proof. Kamil Kielczewski (talk) 20:02, 11 April 2023 (UTC)[reply]

The claims of being especially simple or of being the first non-circular trigonometric proof need secondary sources. We cannot make those claims based only on the original primary publication. —David Eppstein (talk) 20:06, 11 April 2023 (UTC)[reply]

The information about "first non-circular trigonometric proof" was not included into deleted proof (in the same way like "primality" (in some way) of the some other proofs on this page).

Simplicity is obvious because tricky part is only adding zero by: x-(x-y) (and use some old known formulas) - I doubt anyone will describe such obvious things in an article. Kamil Kielczewski (talk) 20:19, 11 April 2023 (UTC)[reply]

This is missing the point. Arguments here for why it's a good proof are not what is needed to justify its inclusion. If nobody has written secondary sources singling it out as a good proof, we cannot include it. —David Eppstein (talk) 20:26, 11 April 2023 (UTC)[reply]

Ok, here is secondary source which mention that this is first trigonometric proof:

"OTHER TRIGONOMETRIC PROOFS ON PYTHAGORAS THEOREM", N. Luzia, 2015, https://arxiv.org/pdf/1502.06628.pdf Kamil Kielczewski (talk) 20:37, 11 April 2023 (UTC)[reply]

That is not reliably published. And it has no depth in its coverage of the Zimba publication. —David Eppstein (talk) 20:54, 11 April 2023 (UTC)[reply]

The Zimba proof relies on the angle-addition formula for sines. However with that formula and γ = α + β, the result is more immediate: one can insert sin α = cos β = a/c, cos α = sin β = b/c, and sin γ = 1 into sin γ = sin α cos β + sin β cos α to give 1 = (a/c)² + (b/c)². —Quantling (talk | contribs) 20:26, 12 April 2023 (UTC)[reply]

Pythagorean theorem dates from more than 1,000 years; trigonometry date from more than 500 years. Since them, hundred of great mathematicians have studied their relationship. So it is very unlikely that something really new can be found on this subject. So, for mentioning Zimba's proof, one requires a secondary source that attests that this is really new. This is really unlikely that this will ever occur for the following reason. The fundational principle on which is based trigonometry is that the trigonometric ratios depend only on one acute angle of a right triangle, and do not depend on the size of the riangle. This is directly used in the proofs of § Proof using similar triangles and § Trigonometric proof using Einstein's construction. Any other trigonometric proof must use this foundational principle. All the proofs suggested in this talk page use this foundational principle and some other trigonometric properties. This makes them definively less interesting and less elegant than the proofs that are already there. So, they have a low encyclopedic value and do not deserve to be mentioned. D.Lazard (talk) 21:39, 12 April 2023 (UTC)[reply]

in your proof you use a,b,c (from geometry object - triangle) - but Zimba use only two arbitrary angles x and y (without involving geometry in direct way like you). Kamil Kielczewski (talk) 09:49, 14 April 2023 (UTC)[reply]

If you don't want a, b and c, the shorter-than-Zimba proof gets even shorter. With the angle-addition formula for sines and γ = α + β, the result is immediate: one can insert cos β = sin α, sin β = cos α, and sin γ = 1 into sin γ = sin α cos β + sin β cos α to give 1 = sin² α + cos² α. —Quantling (talk | contribs) 13:35, 14 April 2023 (UTC)[reply]

${\displaystyle \sin \gamma =1}$ cannot be used because the trigonometric definition of sine as ration of opposite side to hypotenuse does not apply, namely, you cannot have two right angles inside a right triangle! Zimba was careful to note that trigonometric functions of angles ${\displaystyle 0}$ or ${\displaystyle {\frac {\pi }{2}}}$ cannot be directly used. Danko Georgiev (talk) 11:36, 15 April 2023 (UTC)[reply]

In your proof, you assume that sin α = cos β and sin γ = sin(α + β)=1 - I'm not sure that this assumptions are independent of Pythagorean theorem - you also didn't explain where you got these assumptions from? (from geometry - triangle?). Zimba assumptions was weaker than your - he use arbitrary x and y angles and assume only that 0 < y < x < pi/2. (so he did not have to refer to any geometrical figure). This is why Zimba proof is quite interesting and qualitative different from other proofs. Kamil Kielczewski (talk) 14:48, 14 April 2023 (UTC)[reply]

As I see it, the opposite of α is the adjacent of β (and vice-versa) when they are from a right triangle, so sin α = cos β and cos α = sin β follow immediately from the definitions that Zimba gives for sin and cos. Zimba uses that α (well, "x" in his notation) is from a right-triangle when he argues that sin² α + cos² α = 1 leads to (a/c)² + (b/c)². (In contrast, instead of β = π/2 − α, Zimba uses an unrelated angle "y".)

I see that Zimba argues that sin and cos as he defines them are defined only on the open interval (0, π/2), but not at 0 or at π/2. I'm not sure why he couldn't have simply specified the value of those functions at those points and then shown that the subtraction formulas still work when one or more of their inputs are in this expanded domain. Perhaps he considered that less elegant than the approach he did take.

I am curious. Does Zimba claim to be the first to observe that the angle-subtraction formulas for sine and cosine can be proved without assuming the Pythagorean theorem? Does Zimba claim to be the first to observe that the subtraction formulas can be used to prove sin² α + cos² α = 1? Does Zimba claim to be the first to put these two thoughts together? Does Zimba claim that his approach is distinct from previous approaches because he avoided using sin and cos at 0 and π/2? —Quantling (talk | contribs) 16:22, 14 April 2023 (UTC)[reply]

You use sin, cos and γ, α, β with asumption sin α = cos β and sin γ = sin(α + β)=1

He use sin, cos and angles x,y with asumption 0 < y < x < pi/2.

I think that if your sin/cos funtions are the same as Zimba sin/cos functions (at least in (0,pi/2)) then whe shoud not refer to they definitions when we compare proofs - because you both uses same functions.

Zimba only shows that functions sin/cos can be defined independent of Pythagorean theorem, to be sure that using them in proofs is allowed.

But back to the proofs themselves - his proof is just pure symbolic and base only on sin/cos properties (substraction formulas) (which is somehow beautiful), your proof (I supose) need to relate to some triangle.

I'm not sure that Zimba was first - but if not, then should exists similar results before him. But so far I haven't found any Kamil Kielczewski (talk) 17:32, 14 April 2023 (UTC)[reply]

Yes, we'd need a secondary source to make any claim that a proof was 'first'. We can't rely on what editors happen to have found themselves.

MrOllie (talk) 17:53, 14 April 2023 (UTC)[reply

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Yep, but deleted proof (here) not contains information that it was first. Kamil Kielczewski (talk) 18:02, 14 April 2023 (UTC)[reply]

You claimed it was first further up this page. But the text in the article itself presented no indication that it is noteworthy - Which is why it got deleted. Subjective claims about simplicity and simplifying things on the talk page might be a fun diversion, but the only way a mention could stay in the article is with good support from secondary sourcing - and not in the form of self-published arxiv stuff. -

MrOllie (talk) 18:06, 14 April 2023 (UTC)[reply

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MrOllie: Agreed! —Quantling (talk | contribs) 18:15, 14 April 2023 (UTC)[reply

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yep, agree Kamil Kielczewski (talk) 18:37, 14 April 2023 (UTC)[reply]

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MrOllie

I found a solution to this impasse.

Currently in the article in the Algebraic proofs section there is a proof based on this source - so you consider this source to be reliable.

Well, Zimba's proof has also been included in this source which you found reliable (because you allowed this source to be used on this page for many years) here.

In both proofs in this source there is information about who is considered to be the first author of the proof (12th century Hindu mathematician Bhaskara, and Jason Zimba) - although in both proofs on Wikipedia this information is not provided.

Therefore, it can be consistently assumed that information about Zimba's proof is based on reliable sources (unless you have double standards) Kamil Kielczewski (talk) 08:14, 15 April 2023 (UTC)[reply]

The fact that a proof is sourced from a unreliable source does not means that there are not reliable sources for this proof. In fact, the Cut-the-knot page for the algebraic proof refers to several older sources (one is almost 2,000 years old). On the other hand, the Cut-the-knot page for Zimba's article refers only to Zimba's article.

Also, comparing Zimba's proof with that of

Trigonometric ratios

and similarity of right triangles. The latter is simple and direct, while Zimba's proof requires an elaborated geometrical construction and the proof of an auxiliary trigonometric formula.

Also, the last sentence of Zimba's introduction suggest that his aim is to prove the Pythagorean trigonometric identity without using the Pythagorean theorem, rather that proving the Pythagorean theorem without using Pythagorean trigonometric identity. This suggests that his article is not primarily about a proof of the Pythagorean theorem. In any case, § Trigonometric proof using Einstein's construction can be easily modified for proving both simultaneously.

These are technical reason for not including Zimba's proof, but, again, the main reason for not including it is that inclusion requires WP:Notability, and Zimba's article is not notable enough for being mentioned. D.Lazard (talk) 10:02, 15 April 2023 (UTC)[reply]

The definition of trigonometric functions given in

Trigonometric ratios is standard from centuries on, and is independent from Pythagorean theorem. So, Zimba's definition has nothing new. As Pythagorean theorem is about right triangles, it is impossible to provide a proof that does not involve any right triangle. The trigonometric proof given in the article does not require subtraction formula or any other trigonometric identity. D.Lazard (talk) 18:20, 14 April 2023 (UTC)[reply

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I'm surprised by what you write - can you provide a link (or explain it) to a trigonometric proof which not require any other trigonometric identity? Kamil Kielczewski (talk) 18:35, 14 April 2023 (UTC)[reply]

Look at § Trigonometric proof using Einstein's construction. D.Lazard (talk) 10:05, 15 April 2023 (UTC)[reply]

standard from centuries on, – To be precise, this definition dates from about the middle of the 18th century, and became standard somewhere around the middle of the 19th century. –jacobolus (t) 18:46, 14 April 2023 (UTC)[reply]

The section Pythagorean theorem § Relation to the cross product gives true math, but it isn't closely enough related to the Pythagorean theorem. Specifically,

1. Because the sides of length a and b are perpendicular to each other the value of a · b is always zero.
2. The right-hand side, ‖a‖ ‖b‖, doesn't look anything like the Pythagorean theorem's c².

We could improve this by changing occurrences of b to c. In that case the equation (a · c)² + ‖a × c‖² = (‖a‖ ‖c‖)², would be (aa)² + (ab)² = (ac)². I'd make the change to the text, but I don't know how to make the corresponding change to the graphic. Help! —Quantling (talk | contribs) 16:38, 13 April 2023 (UTC)[reply]

I suggest to remove this section. I have never heard of a relationship between Pythagorean theorem and the cross product, and I do not see in the section any indication of such a relationship. D.Lazard (talk) 17:12, 13 April 2023 (UTC)[reply]

Given that the norm of a cross product is a sine times the vector lengths and the dot product is a cosine times the vector lengths, it is pretty straightforward to plug these into a Pythagorean theorem. I'd do it with c instead of b, but otherwise it works. However, big picture, I am neutral as to whether this is sufficiently noteworthy and interesting; if no other editor chimes in, don't let me stop you from deleting the section. (But if there is some support, maybe let's mend it rather than end it.) Thanks —Quantling (talk | contribs) 13:41, 14 April 2023 (UTC)[reply]

There are at least a couple relevant relationships. First, for any two Euclidean vectors ${\displaystyle a}$ and ${\displaystyle b,}$ the geometric product is ${\displaystyle ab=a\wedge b+a\cdot b,}$ and these parts satisfy ${\displaystyle |ab|^{2}=|a\wedge b|^{2}+|a\cdot b|^{2}.}$

Relatedly, if you start with two vectors which are perpendicular ${\displaystyle a\cdot b=0,}$ then you have ${\displaystyle (a+b)^{2}=a^{2}+b^{2}.}$ –jacobolus (t) 18:31, 14 April 2023 (UTC)[reply]

I don't have time now, tho maybe i'll do this myself later. (1) All figures should be numbered, and referred to by number in the text, not just in this section but over the entire article. A good way would be to number sections and do Figure 1-1, 1-2, 2-1, etc. so renumbering does not have to occur as much when edits are done.

(2) There is a two-panel diagram here with an upper and a lower panel. But the text talks about the lower panel first, then the upper, which is confusing. The diagram should be cut in half and made into two, rearranged in the logical order. editeur24 (talk) 14:20, 14 April 2023 (UTC)[reply]

The problem with numbering the figures in semi-popular Wikipedia articles is that the numbering very rarely stays up to date as many Wikipedians make slight changes here and there. It takes someone constantly checking to maintain the numbering. Per the manual of style, sections "[should] not be numbered or lettered as an outline". –jacobolus (t) 19:46, 14 April 2023 (UTC)[reply]

This video by polymathematic demonstrates a trigonometric proof of the Pythagorean theorem recently discovered by Calcea Johnson and Ne'Kiya Jackson, two high school students at St. Mary's Academy in New Orleans, who recently presented it at the (2023?) Spring Southeastern Sectional Meeting of the American Mathematical Society. They used a pure (mostly) trigonometric proof, using what they call a "waffle cone" geometric construction to arrive at the equation a² + b² = 2ab / sin (2a) = c². It would be nice to add this to the article, in the "Trigonometric Proofs" section. (I'm not sure how to present this proof myself.) — Loadmaster (talk) 22:57, 23 April 2023 (UTC)[reply]

See multiple long discussions above, starting at § Proof using trigonometry —David Eppstein (talk) 07:16, 24 April 2023 (UTC)[reply]

The redirect Pythagoras' theorem proof (rational trigonometry) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 April 29 § Pythagoras' theorem proof (rational trigonometry) until a consensus is reached. Jay 💬 06:59, 7 May 2023 (UTC)[reply]

This edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request.

Please, move the first formula on the page to go right after the first paragraph (if you view the page on a mobile device now, you will not see the formula where it should be). Germanivanov0719 (talk) 18:11, 7 May 2023 (UTC)[reply]

I don't understand the request. Can you elaborate? –jacobolus (t) 18:25, 7 May 2023 (UTC)[reply]

I don't understand either. When I view the article on a mobile device (using the Android app on my phone) I do see the formula where it should be, immediately below the first paragraph and above the (minimized) infobox. —David Eppstein (talk) 18:30, 7 May 2023 (UTC)[reply]

I don't know how to minimize the infobox, and I'm using the default theme. Could you try using the DevTools to decrease the view width to see if that breaks it (you need to reload the page after you change the view)? I've tested it on my Android phone and on macOS, both with Chrome, and I have the same problem. Germanivanov0719 (talk) 18:39, 7 May 2023 (UTC)[reply]

Try opening the page on a phone. The first paragraph end with "...often called the Pythagorean equation:", and instead of the the equation you will see this box with information about the theorem. The formula will be below that box. Germanivanov0719 (talk) 18:31, 7 May 2023 (UTC)[reply]

This seems like a Mediawiki problem. The infobox is at the top of the page in the source, and the equation immediately follows the paragraph. I think Mediawiki's mobile view perhaps special-cases the leading image or infobox to move it after the first paragraph? Not sure if there's a good workaround to force the equation to stay with the paragraph. We could perhaps try adding a paragraph break earlier so that the sentence stays with the equation. –jacobolus (t) 18:42, 7 May 2023 (UTC)[reply]

I tried making such a change. We can discuss whether it's worth it to make article content compromises for this, or if there's some other work around, and possibly revert that change. Does that at least fix the problem? –jacobolus (t) 18:50, 7 May 2023 (UTC)[reply]

Yes, now the formula is after the paragraph, which is after the infobox. Germanivanov0719 (talk) 18:55, 7 May 2023 (UTC)[reply]