Talk:Pythagorean theorem

Einstein's proof or Euclid's second proof?

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)

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)

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)

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)

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 $a/b=b/c\implies a/c=ka^{2}/kb^{2},$ though that is somewhat conceptually anachronistic.) –jacobolus (t) 18:13, 10 April 2023 (UTC)

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)

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)

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)

Why Zimba proof was deleted?

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)

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)

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

$\sin \gamma =1$ 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 $0$ or ${\frac {\pi }{2}}$ cannot be directly used. Danko Georgiev (talk) 11:36, 15 April 2023 (UTC)

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)

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)

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)

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)

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

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)

@Kamil Kielczewski: Regarding "his proof is just pure symbolic ... your proof (I supose) need to relate to some triangle." He uses triangles, but I suppose that you mean right triangles. Yes, agreed, he gets all the way to sin² x + cos² x = 1 without referring to a right triangle, though he needs a right triangle for the next step, to get to (a/c)² + (b/c)² = 1. @MrOllie: Agreed! —Quantling (talk | contribs) 18:15, 14 April 2023 (UTC)

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

@D.Lazard @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)

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 proof using Einstein's construction, I cannot see any advantage of Zimba's proof: both use the definition of sine and cosine given in 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)

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)

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)

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

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)

Relation to the cross product

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)

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)

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)

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

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

Algebraic Proofs: edit request to number and rearrange the diagram

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)

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)

New trigonometric proof

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)

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

"Pythagoras' theorem proof (rational trigonometry)" listed at Redirects for discussion

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)

Semi-protected edit request on 7 May 2023

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)

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

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)

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)

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)

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)

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)

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

Reverse mathematics perspective

Seems like everybody has this backwards. The Pythagorian theorem is a generalization of empirical observations, probably going back to ancient monument construction. Observations that all right triangles satisfy the Pythagorean theorem to within precision of the methods available in ancient times, no matter where or when the measurements are made, provides inductive support for the homogeneity, isotropy, and scale-invariance of the world. We can deduce the parallel postulate and other important elements of geometry from the Pythagorean theorem. Doesn't this seem like it makes more sense than trying to empirically verify the parallel-postulate? And at very large scales empirical support for the Pythagorean theorem fails, leading naturally to other geometries. Maybe somebody with greater wiki expertise could add a section on reverse mathematics atleast mentioning this perspective. The article introduction asserts that there are many ways to "prove" the Pythagorean theorem, but gives no clear acknowledgement of the parallel postulate or alternatives upon which proofs should be critiqued. Uscitizenjason (talk) 19:13, 8 January 2024 (UTC)

If you want this perspective to be represented in the article, you are going to need to find published and scholarly sources that express the same sentiments. We cannot add material based purely on the musings of random Wikipedia editors. —David Eppstein (talk) 21:07, 8 January 2024 (UTC)

Well, the idea of using axioms that were empirically motivatable was part of the spirit of Birkhoff's axioms of geometry, although he chose to use similar triangles rather than the Pythagorean theorem. Uscitizenjason (talk) 18:57, 12 January 2024 (UTC)

I'd say the "idea of using axioms that were empirically motivatable" was most of the spirit of Euclid's axioms (and various alternatives over the following centuries).

In any event, it is certainly the case that you could reshuffle your set of axioms to include the Pythagorean relation, if you wanted to. I'm not sure to what extent, if any, discussing this point is super useful in the context this page. –jacobolus (t) 21:02, 12 January 2024 (UTC)

Until the middle of the 19th century, all axioms of mathematics were abstractions of empirical experiments. You are talking of the relationship between the parallel postulate and the Pythagorean theorem. It is true that for proving the Pythagorean theorem, one needs the parallel postulate or something equivalent. But the converse is not true, since the Pythagorean requires a notion of distance. In particular, in an affine space, the parallel postulate is verified, but there is no notion of right angle.

I understand your "reverse perspective" as the study of the axioms that are needed for proving some theorems. Emil Artin's book Geometric algebra is a rather complete study of this kind of questions. D.Lazard (talk) 20:39, 12 January 2024 (UTC)

The page on the parallel postulate claims that the Pythagorean theorem is equivalent to the fifth postulate. I've added a brief blurb to this effect in the Pythagorean theorem article. —Quantling (talk | contribs) 20:49, 12 January 2024 (UTC)

That's in conjunction with postulate 3 saying you can draw circles (and various other assumptions left unstated by the Elements, such as that a circle intersects every line through its center). –jacobolus (t) 21:05, 12 January 2024 (UTC)

Simple algebraic proof using similar triangles

Looking at the hypotenuse and height of the three similar triangles, we can write the following products and ratios relationships, then multiply them:

a·a' +  b·b'  =  c·c'    (products = 2 x areas)

a/a' =  b/b'  =  c/c'    (ratios)

$a^{2}+b^{2}=c^{2}$

Dividing them naturally also gives us:  a'² + b'² = c'²

[note to self: add reliable sources here]

Weallwiki (talk) 16:20, 7 March 2024 (UTC)

This page doesn't need more proofs, unless they are (a) published in reliable sources, and (b) in some way particularly notable or interesting, as described in reliable sources. We already have more than enough proofs to make the general point that the possible list of proofs is endless. With that said though, this is a fine proof. Nice work. If you can find some website that attempts to comprehensively list as many proofs as possible, you could submit this there. (Unfortunately Alexander Bogomolny died a few years ago, so I don't think cut-the-knot.org/pythagoras/ is taking new submissions.) –jacobolus (t) 16:57, 7 March 2024 (UTC)

This is fairly simple, so I like that. I see that the triangles are similar, but we'd want to explain that. I do hope you find a reliable source that shows that this is sufficiently notable. —Quantling (talk | contribs) 17:41, 7 March 2024 (UTC)

Thanks for the great feedback so far, will look into making these improvements. Weallwiki (talk) 21:21, 7 March 2024 (UTC)

Create an article for proof of Pythagorean theorem's only

Note that the article Pythagorean theorem should focus on explaining the Pythagorean theorem. However, at this point, the article also contains lots of proof of this theorem. In that case, should both sections Pythagorean theorem#Proofs using constructed squares and Pythagorean theorem#Other proofs of the theorem be split into the article Proofs of Pythagorean theorem? The fact I have discussed in the WT:GAN, and IMO this regards GACR2a and GACR3b. Dedhert.Jr (talk) 10:20, 12 March 2024 (UTC)

I have no clear opinion whether the article must be split. However, here are some comments.

• The sections on proofs are presently in the middle of the explanations of the theorem, its consequences and its applications. Readers interested in these aspects of the theorem have thus to skip a wall of text that can be interesting in the whole for very few readers only. So, an immediate very useful action would be to move these sections toward the end of the article, possibly with a link in the lead.
• The sections on proofs require to be restructured and largely rewritten. Presently they appear as an WP:indiscriminate list, and, often, the headings do not give the needed information on the specifity of the proof method.
• It seems that the main reason for a split is that, without a split, much more work is needed to reach the good-article status. I do not know whether tis is a good reason for a split

D.Lazard (talk) 13:12, 12 March 2024 (UTC)

I don't think a split is necessary. There's not so much material here that it can't fit in a single article, and proofs are obviously one of the main things to discuss about a theorem. The sections on proofs should definitely be better organized for narrative flow. This kind of list that slowly accretes inconsistent items without curation is pretty common among popular older pages. I just tried to do some cleanup on the somewhat similar list of derivations at quadratic formula.

I don't think making readers skim past roughly the current quantity of text about various proofs is necessarily a problem – the proofs are important and insightful – but we should make some effort to make reading through the text pleasant and comprehensible. More important in my opinion is to find clear sources for every proof, ideally mention who first made each proof and link to the original, make the formatting and illustrations a bit more orderly and maybe more consistent in style.

A couple more notes: Even if the article is split at least 5–6 different proofs should be covered in detail on the main page, taking roughly as much space they currently take. I'm concerned that an explicit article about proofs would become an indiscriminate grab-bag of mediocre crap, and it would be harder to push back against adding this or that arbitrary proof that anyone wants to include. –jacobolus (t) 14:39, 12 March 2024 (UTC)

I agree in spirit that some proofs should remain behind, with a pointer to the (new) main article that has those and additional proofs. I might haggle over whether it should be 5–6 vs. 2–3 that survive in the present article, but that's just details.

Yes, the new article could become a grab-bag, but I think that that is okay. If the user has come looking for proofs, let's give them proofs. We'll have some minimum standards of course, but we can make the threshold a little lower than it is for proofs that are presently in this article. —Quantling (talk | contribs) 16:27, 12 March 2024 (UTC)

I think the grab-bag articles should generally be avoided where it's relatively straightforward to do so. They typically end up turning into substantially useless unreadable sludge. In the case where there is some important reference material involved, e.g. list of trigonometric identities, some readers might be willing to wade through that to find a point they are looking for (though I question how many), but for something like a list of proofs this doesn't seem that valuable to me. I would instead just direct readers to cut-the-knot.org/pythagoras, Loomis (1968) The Pythagorean Proposition (alternate scan), etc. –jacobolus (t) 16:35, 12 March 2024 (UTC)

If the alternative way is keeping them in the article, the scenario I imagined would probably restructure sections in which the article presents the statement of theorem and its converse firstly and then a single proof of the theorem, and add the link, redirecting the latter section. Dedhert.Jr (talk) 12:27, 13 March 2024 (UTC)

Support split: I would like to see that split. I think that readers who are looking for multiple proofs can be substantially different from readers who are looking to learn non-proof aspects of the Pythagorean theorem. I think that fully supporting both goals, now and into the future, will make a single article too long and too hard to navigate. —Quantling (talk | contribs) 15:26, 12 March 2024 (UTC)

Comment: The new article's title might be Proofs of the Pythagorean theorem. —Quantling (talk | contribs) 16:33, 12 March 2024 (UTC)

I am inclined against this, on general following-the-sources grounds. In my experience, the texts that cover the Pythagorean theorem at an introductory level don't just apply it; they prove it in one or more ways. We'd be the oddballs if we separated the proofs out entirely. Doing mundane cleanup and readability-improvement work on the material currently in the article seems more important. XOR'easter (talk) 18:49, 12 March 2024 (UTC)

I'm inclined against this on somewhat different grounds: having an article specifically devoted to collecting proofs of the theorem seems likely to grow into a huge indiscriminate collection of proofs, something that I do not think would make for a good encyclopedia article. It would be a cruft magnet. That sort of thing is only marginally effective at keeping the cruft out of the main article and instead encourages the accumulation of more cruft. Instead, keeping it only in this one article maintains the pressure to stay at roughly the amount of content that we already have: a properly sourced statement that there are huge numbers of proofs that you can find in certain books, and a small (and I hope carefully-curated) selection of proofs. —David Eppstein (talk) 19:41, 12 March 2024 (UTC)

I remember that the list of all proofs may be suggested to relocate them into the WikiBooks. If this is a good idea, maybe we can add the link in the external link. However, I prefer to hear from others. Dedhert.Jr (talk) 12:36, 13 March 2024 (UTC)

Support split. As others here will know, there has been much discussion about when and whether proofs should be included in mathematics articles (see for example: Wikipedia talk:WikiProject Mathematics/Proofs, Wikipedia talk:WikiProject Mathematics/Proofs/Archive 1, Wikipedia talk:WikiProject Mathematics/Proofs/Archive 1, as well as this search list). I've been involved in many of these, and I believe the general consensus has been that most proofs have little encyclopedic value. But some do (e.g. the irrationality of the Square root of 2, Cantor's diagonal argument, Gödel's incompleteness theorems), and I also believe that some proofs of the Pythagorean theorem do too, but certainly not all (or at least not in this article). However this theorem is unique in that there have been so many proofs discovered (or created ;-)), so that, to me, an article devoted to them seems warranted. Of course, as jacobolus points out above we need reliable sources for every proof we publish, and it seems to me that rigid enforcement of this would deal with the "cruft" problem. Paul August ☎ 13:43, 13 March 2024 (UTC)

Proofs aren't particularly helpful for validating most statement in most encyclopedia articles. In articles about a broad topic or field of study it's sometimes worth having a short proof or two as illustrative examples rather than as validation for claims made. But in an article about a theorem a proof or proofs are obviously directly relevant. Indeed I would hope every article about a theorem should include at least some kind of proof sketch or motivating idea, and articles about theorems famous for their multiple proofs should describe or include the most noteworthy ones (to the extent practical; obviously some proofs are extremely long or technical). Clearly all (infinitely many) proofs of the Pythagorean theorem can't be in scope here, but the proofs can be categorized into 4–5 broad groups, and 1–3 notable examples from each group should be included on this page, irrespective of what material is included on other articles. Many are quite short or can be expressed pictorially. –jacobolus (t) 15:04, 13 March 2024 (UTC)

• Support split per above discussion.

Youprayteas ^{talk/contribs} 13:28, 14 April 2024 (UTC)

I can see an interest of (some) readers to have comprehensive collection of proofs, which doesn't fit into this article. But imho Wikipedia is not the appropriate place for that, there are other options within in Wikimedia to provide such a collection to readers. One could integrate it into existing Wikibook projects for proofs or set up a dedicated Wikibook project just for this collection. As an external option there is the ProofWiki project. Our article should offer links to such collections in the external links section.--Kmhkmh (talk) 23:08, 13 March 2024 (UTC)

... there are other options within in Wikimedia to provide such a collection to readers.

This may be the way to go. I am not familiar with these other ways. For example, I consider Wikipedia to be fairly reliable because there are many good editors keeping an eye out for quality; are these other options as reliable in practice and by reputation? Because, if not, I'd like there to be a reliable collection in Wikipedia itself in Proofs of the Pythagorean theorem. —Quantling (talk | contribs) 14:32, 14 March 2024 (UTC)

Yes but there is the rub. Wikipedia is fairly reliable and you have good editors keeping an eye because we restrict our content. That exactly a reason to avoid long proofs or list of long proves as their verification takes more time/resources and they are less likely to be checked in detail by other editors.--Kmhkmh (talk) 05:26, 16 March 2024 (UTC)