Talk:Ptolemy's theorem

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Am I the only one who finds the formatting on this page screwy in Firefox?

Probably got fixed some time in '05.

"Any square can be inscribed in a circle whose center is the barycenter of the square."

Why not simply the center of the square? If barycenter is indeed necessary, then it should be made a link to the barycenter article - the term is not common enough to be generally understood in my opinion.

I've moved this from Ptolemaios's theorem to its much better known name, Ptolemy's theorem. Although I'm Greek myself, there's no real reason to use obscure Greek versions of names when there is a much more widely used form that's standard in English. --Delirium 08:43, 10 November 2005 (UTC)[reply]

Why not introduce the theorem with the much simpler ABCD notation used in the proof and its figure, instead of the complicated-looking point–subscript notation? OK if I change it?

Current way:

If the quadrilateral is given by its four vertices P₁, P₂, P₃ and P₄, then the relation is

${\displaystyle {\overline {P_{1}P_{3}}}\cdot {\overline {P_{2}P_{4}}}={\overline {P_{1}P_{2}}}\cdot {\overline {P_{3}P_{4}}}+{\overline {P_{1}P_{4}}}\cdot {\overline {P_{2}P_{3}}}}$

Here ${\displaystyle {\overline {P_{i}P_{i+1}}}={\overline {P_{i+1}P_{i}}}}$ for ${\displaystyle i=1,\ldots ,4}$ denote the lengths of the four sides of the quadrilateral (with indices taken modulo 4), the two diagonals are then ${\displaystyle {\overline {P_{1}P_{3}}}}$ and ${\displaystyle {\overline {P_{2}P_{4}}}}$.

Proposed new way:

If the quadrilateral is given by its four vertices A, B, C, and D in order, then the theorem states that:

${\displaystyle {\overline {AC}}\cdot {\overline {BD}}={\overline {AB}}\cdot {\overline {CD}}+{\overline {BC}}\cdot {\overline {DA}}}$

where the overbar denotes the lengths of the line segments between the named vertices.

And of course it could use an introductory illustration. Dicklyon 18:59, 28 August 2006 (UTC)[reply]

Hearing no objection, I'll go ahead. And I added a figure for the golden ratio application, too. Dicklyon 02:15, 29 August 2006 (UTC)[reply]

And I mangled the illustration to make one for the lead. These are the first two SVGs I've ever made, so I'm still running a bit clueless. Please feel free to improve, replace, or delete. Dicklyon 02:48, 29 August 2006 (UTC)[reply]

When someone suggests a merge, it would be proper to open the merge discussion with the reasoning. But, since Michael Kinyon didn't, I will. I was not previously aware of Ptolemy's inequality, but since it has a tiny article and explains that equality is achieved only in the case the corresponds to Ptolemy's theorem, it seems like a no-brainer to merge it into here. Perhaps it's all part of the same theorem originally? Or posed as a corrolary? We should find out, to determine HOW to merge it exactly. Dicklyon 03:04, 28 September 2006 (UTC)[reply]

Sorry I didn't say something; after posting the merge tags, I got a bit busy in real life, and then it slipped my mind. Thanks for the Talk page reminder. In any case, I too consider this a no-brainer, and in fact, it is normally the sort of bold merge an editor does without bothering to post tags. However, a couple of folks other than myself, namely Bh3u4m and Charles Matthews, did some recent editing on Ptolemy's inequality, so I thought it better to tag the articles. The other difficulty is that Ptolemy's theorem is quite dense already and has multiple proofs in it, so I am not sure exactly how to work the inequality case into it. I was hoping that those of you who watch this page more closely than I do would have some ideas. I am not sure about the history of the two results. Michael Kinyon 12:55, 28 September 2006 (UTC)[reply]

PlanetMath calls Ptolemy's theorem the equality [1], and has and article about Ptolemy's inequality proof [2]. I would therefore conclude that the theorem is a special case of the inequality (when equality occurs), thus I would leave two distinct articles, specifying this feature. Bh3u4m 13:12, 28 September 2006 (UTC)[reply]

Most sources treat them together. Here's a book that treats the inequality as a strengthening of the converse of the theorem: [3] Very few books name the "Ptolemy's inequality;" most just include the inequality as an extension or corrolary of the theorem, but leave it unnamed, it appears. Dicklyon 13:58, 28 September 2006 (UTC)[reply]

That's a nice source. I think we could work the inequality into the Preview and into the Geometric Proof pretty easily. I am not enamored of the other proofs, but I suspect those who are could work out how to rewrite them appropriately. Michael Kinyon 15:32, 28 September 2006 (UTC)[reply]

If I'm not wrong, PlanetMath is published under the GFDL, so it would be possible to simply copy their proof (this has to be verified) Bh3u4m 16:16, 28 September 2006 (UTC)[reply]

Nice work with the merge, Dicklyon. Michael Kinyon 23:45, 7 October 2006 (UTC)[reply]

And thanks for fixing my typos, Michael Kinyon. Dicklyon 00:01, 8 October 2006 (UTC)[reply]

I unmerged and re-expanded the inequality article. See Ptolemy's inequality. —David Eppstein (talk) 05:50, 6 June 2016 (UTC)[reply]

I just tagged this article for cleanup. I am concerned about the emphasis on "ancient magi" in a mathematics article, and while I understand very well the need for proofs in mathematics I don't see the need for four different long tedious derivations in this article. In addition, the content needs to be sourced: where do these proofs come from, where do the corollaries come from, etc.? There is a little of this already, but far too little. —David Eppstein (talk) 17:03, 10 April 2008 (UTC)[reply]

Agreed. I removed the unsourced tag since the artcile no longer fit the description in the tag. --Pleasantville (talk) 17:20, 10 April 2008 (UTC)[reply]

Can I suggest moving most of the proofs away to a 'stub' article which can be referenced as 'Other proofs'? The main article can just have the classical geometric proof - which in any case is by far the most elegant! I will alter "ancient magi" to "ancient geometers" and I hope that is more appropriate to a mathematical article which nonetheless needs to acknowledge its historical roots. Neil Parker (talk) 08:04, 25 April 2008 (UTC)[reply]

The proofs are not there to establish the truth of the theorem (that's not wikipedia's job). They are there to show the connections of the theorem to other statements in mathematics. Therefore, all significant proofs should be in the article.--345Kai (talk) 22:39, 15 May 2008 (UTC)[reply]

David - I posted a response re corollaries on your talk page. There doesn't seem much else we can do about the proofs unless the authors aggree to move them all off to a 'further reading' section or something of that nature. So what next - I would like to work towards making this a 'model article' since the theorem is of considerable historical importance and indeed is the basis of Copernicus' table of chords which is used in the trigonometric calculations underpinning his revolutionary helio-centric model. Neil Parker (talk) 15:50, 27 June 2008 (UTC)[reply]

Why can't we do something about the proofs? They don't add much to the encyclopedic nature of the article; it would be more sensible to describe what each proof connects to that makes it interesting, rather than do the proof out in detail, and to cite a source for the proof itself. Dicklyon (talk) 16:00, 27 June 2008 (UTC)[reply]

I would suggest definitely keep the original geometric proof (Theorema Secundum in Book 1 of 'De Revolutionibus Orbirum Coelestium') since it is very much a part of the theorem's very ancient historical roots. Otherwise I can only speak for Proof 3 which was submitted in order to present Ptolemy's Theorem in modern trigonometric form and hence to derive the corollaries also in modern trig form (there is nothing new in any of these since in essence they were all known to Copernicus and Ptolemy before him). I don't have source material other than Hawking's book but - with a theorem as old as this - there can be nothing new under the sun so it would be foolish to claim anything original. What I have submitted is relatively obvious and perhaps simply not considered original enough for anyone to have produced a separate citeable reference. If there is such I am quite happy for the proof to be replaced with said reference. Otherwise please advise on a suitable journal and I will send them an article which probably won't be accepted on account of plagiarising an existing Wikipedia article! Neil Parker (talk) 07:56, 29 June 2008 (UTC)[reply]

Generally an extensive number of proofs (or even proofs at all) are not needed in articles regarding a mathematical theorems. What is required instead is reputable sources/references (which may contain the proofs). However since Wikipedia is not paper it doesn't necessarily harm if proofs are added, as long as they don't destroy the structure or readability of the article, they might be ok (and useful to some readers). But it also should be kept in mind, that an extensive amount of proofs is less likely to proofread by other authors and hence can present a potential quality problem for Wikipedia. Personally I'd support the idea of moving the proofs to a separate lemma. Note also that there is wikibook project which collects proof for math theorem ([4]), which might be another good place to put such an extensive list of proofs.--Kmhkmh (talk) 13:18, 9 January 2009 (UTC)[reply]

perhaps I missed it, but for the first figure with the quadrilateral, is the diameter of the circle known, and if so, what is it?--Billymac00 (talk) 17:42, 24 June 2008 (UTC)[reply]

Interesting question - cosine of angle ${\displaystyle \theta }$ between (for example) sides S₃ and S₄ of a cyclic quadrilateral is given by the following formula:

${\displaystyle {\frac {(S_{4})^{2}+(S_{3})^{2}-(S_{1})^{2}-(S_{2})^{2}}{2(S_{1}.S_{2}+S_{3}.S_{4})}}}$.

Once this angle is known, the diagonal d subtending it can be determined using the cos rule and thereafter the ratio ${\displaystyle {\frac {d}{\sin \theta }}}$ gives the circle diameter.

There must be a more elegant way!?

Neil Parker (talk) 18:18, 15 July 2008 (UTC)[reply]

In this diff, User:SmackBot has inserted "totally disputed section" into some math equations, apparently because it confused something in the equation with a tag template and didn't notice it was in a math context. It's too late to "undo", so it has to be repaired by hand. I've stopped the bot and left a note at User Talk:SmackBot, but so far it's ignored. Dicklyon (talk) 14:59, 10 October 2008 (UTC)[reply]

I would like to propose the following re-organisation of the Examples section: See contents section above for how it will look:

Neil Parker (talk) 15:54, 14 April 2009 (UTC)[reply]

I find the notation in this article confusing. I've not seen a horizontal over-bar used to represent the magnitude of a vector before. It's also very close to the over-arrow notation used in Euclidean vector. When I see the following, I think it's using dot products.

${\displaystyle {\overline {AC}}\cdot {\overline {BD}}={\overline {AB}}\cdot {\overline {CD}}+{\overline {BC}}\cdot {\overline {AD}}}$

I suggest using one of:

${\displaystyle AC\cdot BD=AB\cdot CD+BC\cdot AD}$

or

${\displaystyle AC\times BD=AB\times CD+BC\times AD}$

Nturton (talk) 11:32, 7 May 2009 (UTC)[reply]

Your alternative suggestions make AB etc look like products of numbers A and B, even more confusing. As it is, the overbar prevents the product interpretation, and is clearly explained immediately after it is used. If you read the explanation, you will see by the way that it is not intended as the magnitude of a vector but as the length of a line segment. That is, it is intended to convey a geometric rather than linear-algebraic idea. —David Eppstein (talk) 15:19, 7 May 2009 (UTC)[reply]

Why not

${\displaystyle |AC|\cdot |BD|=|AB|\cdot |CD|+|BC|\cdot |AD|}$

Bo Jacoby (talk) 22:10, 27 March 2010 (UTC).[reply]

In geometry, the form, ${\displaystyle AC\cdot BD=AB\cdot CD+BC\cdot AD}$, has been widely used and no confusion arises in the given context. --Roland 23:17, 15 February 2015 (UTC)

To confuse furhermore, i personnally use he notation ${\displaystyle ab+cd=ef}$Catharaxie (talk) 08:48, 5 August 2019 (UTC)[reply]

I have worked on the proposed re-organisation above. Request comments on the draft version which can be viewed here.

Eventually I just posted the changes - hope that's ok ?

Neil Parker (talk) 12:47, 20 October 2010 (UTC)[reply]

Re this proof. The more of these I see, the more impressive is the simple elegance of the original geometric proof! It has about it the indelible stamp of some ancient genius who may predate even Ptolemy himself.

Neil Parker (talk) 13:45, 30 November 2010 (UTC)[reply]

Ptolemy's law for a collinear 4-tuple has a stronger version with signed lengths, useful in projective geometry. Tkuvho (talk) 08:42, 15 December 2010 (UTC)[reply]

What exactly is meant by "Complement of Pentagon Chord"? Angles have complements. Do chords have complements??--WickerGuy (talk) 18:43, 2 March 2012 (UTC)[reply]

That whole section is rather dubious should be removed or at least rewritten.--RDBury (talk) 03:13, 5 March 2012 (UTC)[reply]

The figure and terminology came in here, and the section division later by the same editor. Ask him. Dicklyon (talk) 05:19, 5 March 2012 (UTC)[reply]

The work of Ptolemy followed by Copernicus seems (at least to my admittedly modest understanding) to revolve around the geometry of chords subtending angles. Since a diameter subtends a right angle, the chords which form the right angle subtend angles which are complementary. Well I suppose I could write - as did Copernicus - "the chord which subtends the rest of the semicircle" but that would be a little long winded I think! So "complementary chords" in this context are simply those which form a right angle in the semicircle and therefore subtend complementary angles at the circumference.

In making sense of chord based geometry/trigonometry, one must constantly keep in mind that any arc/chord subtends an angle at the circumference and twice that angle at the centre. In the Copernican "Porism" the side of a decagon (length 61803 parts) subtends 36 degrees at the centre and the remaining 144 degrees of the semicircle is subtended by a chord (length 190211 parts) which is determined from the side of the decagon with the inscribing circle diameter being assigned 200000 parts. At the circumference the angles subtended by these chords are 18 degrees and 72 degrees respectively and they are complementary. The question is how can we economically describe the all important chord subtending 72 degrees - the meaning we need to convey is "the chord which subtends the complement of the angle which is subtended by the side of the decagon" ?. The same applies to the side of the pentagon subtending 36 degrees at the circumference and "the chord which subtends the complement of the angle which is subtended by the side of the pentagon". Please do suggest alternative terminology which better conveys the historical context.

PS: Please could RDBury be a little more specific as to what he considers "dubious" about this section.

Neil Parker (talk) 08:00, 8 August 2012 (UTC)[reply]

From David Eppstein

→‎Examples: the later examples in this section are unnecessarily detailed *as examples*, badly sourced (the sources concern the values themselves and not their derivation via Ptolemy), and unencyclopedically written (too discursive) — remove them

The examples were included because of their historical importance in the determination of tables of chords (effectively sine values) by early astronomers such as Ptolemy and - following in his tracks - Copernicus. I disagree entirely that they were "unnecessarily detailed" - they just applied the theorem as it relates to various chords of pentagon and decagon. Along with chords of triangles and squares, these were the starting points whence further chords were calculated largely via continued application of Ptolemy's Theorem and/or circle geometry theorems as detailed in De Revolutionibus Orbium Coelestium. Mr Eppstein might benefit greatly from a careful study of the latter.

The language can be tidied up but I do not think Mr Eppstein is doing anyone any favours by removing them altogether. I would appreciate it if he rather engaged in a discussion as to how these examples could be tidied up whilst not detracting from their historical value. I do not think it is out of order to demonstrate (for example) the result of Euclid XIII 10 arising via a straightforward application of Ptolemy's theorem. "Source" in this case is very simple - it is Ptolemy's theorem as is the case for all the other "allowed" examples such as Pythagoras' theorem, law of cosines etc.

--Neil Parker (talk) 16:52, 20 May 2016 (UTC)[reply]

As long as their are readded with proper sourcing there is probably no issue. --Kmhkmh (talk) 17:49, 20 May 2016 (UTC)[reply]

Well, also with more encyclopedic writing and less woo. "illuminates another means whereby the ancients may have reached an understanding" — really? —David Eppstein (talk) 18:11, 20 May 2016 (UTC)[reply]

Well hopefully such formulation and style issues will vanish, when the text is based on proper source.--Kmhkmh (talk) 19:42, 20 May 2016 (UTC)[reply]

I indicated clearly enough above - the language can be tidied up. That's a fair point. However the fact remains that , based on a relatively minor stylistic issue, Mr Eppstein has dumped a whole lot of historically important material much of it sourced from one of the all time most important of scientific works namely "De Revolutionibus Orbium Coelestium". In general I'm not into 'woo' Mr Eppstein - what I have (or had) supplied in the article is fundamentally sound Mathematics. If you can't be bothered to read and understand the relevant historical text, that's your problem and not mine.

It was not unreasonable to observe that what I could derive using Ptolemy's theorem , could also have been derived by the likes of Copernicus and his predecessors. Your are welcome to suggest a more 'encylopedic' turn of phrase for that observation than the way I chose to express it. Neil Parker (talk) 16:38, 24 October 2016 (UTC)[reply]

The proof includes that AD=2Rsin(alpha+beta+gamma) I am not a mathematician, and I find it difficult to understand why. AD=2Rsin(2pi-(alpha+beta+gamma)) makes sense to me, because the sum of all 4 angles is 2pi. Any comment is appreciated. — Preceding unsigned comment added by 50.193.173.17 (talk) 16:07, 18 November 2016 (UTC)[reply]

Slightly modified the text by introducing an intermediate ${\displaystyle \delta }$. Actually, the sum of the 4 inscribed angles subtended by the 4 sides of the quadrilateral is ${\displaystyle \pi }$, not ${\displaystyle 2\pi }$. --Roland (talk) 05:14, 25 April 2017 (UTC)[reply]

There's a very elegant and short proof using complex numbers. It's really too bad that it isn't on this page. But an earlier version of this page had a proof using complex numbers, and it was deleted. Puffysphere (talk) 16:13, 26 March 2017 (UTC)[reply]

Yes, so "elegant" and "short" that it went on and on for two and a half screens full of equations in this version. And had no sources. Do you have a good source for a version of this that is actually elegant and short? —David Eppstein (talk) 16:44, 26 March 2017 (UTC)[reply]

Suppose A, B, C and D are the vertices of a quadrilateral, going in order around a circle of radius 1, centered at the origin. Then set

A = e^ia

B = e^ib

C = e^ic

D = e^id

We have the identity

(B−A)(D−C) + (C−B)(D−A) = (C−A)(D−B).

Then (B−A)(D−C) = (ie^i(a+b)/2 |AB|)(ie^i(c+d)/2|CD|) = −e^i(a+b+c+d)/2|AB||CD|,

and (C−B)(D−A) = (ie^i(b+c)/2|BC|)(ie^i(a+d)/2|AD|)= −e^i(a+b+c+d)/2|BC||AD|.

Since arg((B−A)(D−C)) = arg((C−B)(D−A)), their magnitudes simply add, and the theorem is proved.

This approach can also be used to show the converse: that if ABCD is a quadrilateral such that |AB||CD| + |BC||AD| = |AC||BD|, then A,B,C,D are points on a circle, arranged in that order on the circle. Puffysphere (talk) 17:08, 26 March 2017 (UTC)[reply]

Respected sir,

i am an independent researcher who works on the new proofs of classical proofs of euclidean geometry, recently i worked on the new proof of Ptolemys theorem, i actually proved this theorem in some elegant way which was published in reputed journals, i am attaching the details of those, please check once try to add these as the reference list for the Wikipedia article of ptolemys theorem, so that every one go through it,  i proved some new generalisation of this theorem in these articles, so once you go through it and verify accordingly.

http://article.sciencepublishinggroup.com/html/10.11648.j.mcs.20160104.14.html

http://jcgeometry.org/Articles/Volume4/Krishna.pdf

please respond accordingly.

Interesting but the problem would be including all those lemmas. I think the original geometrical proof still stands out on account of its simplicity and reliance on nothing other than basic circle geometry and the properties of similar triangles.

--Neil Parker (talk) 10:09, 3 April 2018 (UTC)[reply]

From David Eppstein

→‎Examples: the later examples in this section are unnecessarily detailed *as examples*, badly sourced (the sources concern the values themselves and not their derivation via Ptolemy), and unencyclopedically written (too discursive) — remove them

Probably we could have done something about the "unnecessary detail" if that was a problem. "Unencyclopedically written" sounds a little subjective as a reason for deletion. We are left with "badly sourced". The material I used was sourced from "De Revolutionibus Orbium Coelestium". Is that a "bad source" or what exactly is the problem ? I can't make any sense of "the sources concern the values themselves and not their derivation via Ptolemy".

I strongly object to sweeping deletions without any attempt to engage serious authors. I formally request Mr Eppstein to engage in the appropriate forum which is this talk page.

--Neil Parker (talk) 09:26, 3 April 2018 (UTC)[reply]

Well, I'd rather see those as applications in elementary geometry than "examples". Strictly speaking to prove their notability for inclusion one might require more extensive sourcing, but given that this is an article in elementary geometry and the examples are straight forward and the article is not suffering from too much content, I don't think being overly formal really serves WP in this case. In other words i'd leave them in the article.--Kmhkmh (talk) 14:38, 3 April 2018 (UTC)[reply]

P.S. The comment above refers to current state of the article and the examples section. If you are talking about the 2014 deletions however I largely agree with Eppstein, so please do not readd them.

One can probably argue that communication 4 years ago probably wasn't optimal and maybe the discussion page should have been used first, but it is what it is now. There is no obligation to engage in lengthy discussion for an edit that "obviously" seems justified. Please note in that regard that proofs in encyclopedic math articles are not needed and are optional at best. Proofs maybe added in individual cases if they are not too lengthy or technical, don't bloat article or somewhat impair it's overall readability. The current proof section is more than sufficient and the material back then added another 20k, which imho was too much.--Kmhkmh (talk) 15:06, 3 April 2018 (UTC)[reply]

What do you "agree" with - can you be specific about which of the "obvious" 2014 deletions you are referring to ? Apart from further interesting examples/applications of the theorem, there were wholesale deletions of relevant historical material which I researched directly from "De Revolutionibus Orbium Coelestium". My sense is that neither you nor Mr Eppstein have a clue about that - hence the nonsensical statement "the sources concern the values themselves and not their derivation via Ptolemy". Everything - from Thm 2 (the theory itself) through Thm 5 - consists of applications/derivations using Ptolemy's theorem. Mr Eppstein's solution to things he didn't understand appears to have been "delete all". Your remarks display a similar lack of discernment. Finally I would observe that communication via the talk pages was just as possible 4 years ago as it is now. I at least am engaging on the talk pages before making any changes - a courtesy that was not afforded to me then. Neil Parker (talk) 12:37, 24 October 2019 (UTC)[reply]

The state of the article prior to the deletions was an unreadable mass of section after section giving examples, proofs, and corollaries, with no attempt to fit them into any clear structure rather than just concatenating them one after another, no attempt at explaining what insight the proof was intended to convey beyond the bare fact of the truth of the theorem (for which a single proof would suffice), and many of them with no sources at all. I stand by my deletions. As for your ad hominem argumentation here, please read WP:AGF and WP:CIVIL. Finally, if you really insist on addressing me as [title] [surname] in the context of my professional expertise, the correct title to use is either "Dr." or "Prof.". Or, you could just use my first name instead. —David Eppstein (talk) 17:32, 26 October 2019 (UTC)[reply]

The article at present covers the application of Ptolemy's Theorem (PT) to the formulas for sine (a + b), sine (a - b), and cosine (a + b), but not the remaining case of cosine (a - b). I was specifically looking for this, so I was disappointed not to find it. I have not found a strictly geometric proof of the formula, using PT, anywhere else. Most sources either just say 'it can also be proved', or give an algebraic proof by writing cos (a - b) as sine [(90 - a) + b] and using the formula for the sum of sines. This works, but it would be nice to have a direct geometric application of PT to the cosine-difference case. (There are geometric proofs which don't use PT, but these don't serve the present purpose.) Eventually I found a geometric proof of my own, as follows. (Sorry, no diagram.) Suppose a and b are angles greater than 0 and less than 90 degrees, with a greater than b. Draw a circle with centre O and diameter AOC = 1. Inscribe a quadrilateral in the circle with its vertices A, B, C, D in that order on the circumference, and constructed so that angle BCA = a, and angle CAD = b. Draw a diameter DOE meeting the circumference at E. Join A to E, E to B, and B to D. From these constructions and standard theorems it follows that ED = AC = 1; OA = OD; angles ABC, EBD, CDA and EAD are right; angle BCA = BDA = a; angle ODA = OAD = b, and angle BDE = BDA - ODA = (a - b). Therefore cos (a - b) = BD (in triangle EBD with hypoteneuse ED = 1.) By Ptolemy's Theorem, in the quad. ABCD, BD.AC = BC.AD + AB.CD. But BD = cos (a - b) (proved above), AC = 1, BC = cos a (in triangle ABC), AD = cos b (in triangle ACD), AB = sin a (in triangle ABC) and CD = sin b (in triangle ACD). Therefore cos (a - b) = (cos a)(cos b) + (sin a)(sin b). Q.E.D. I was pleased to find this proof, but I don't suppose for a moment I am the first. If anyone knows a source containing an equivalent proof and meeting Wiki's citation standards, it would be useful to add a reference to it.109.149.185.151 (talk) 10:27, 22 October 2019 (UTC)[reply]

Theorem 3 (Theorema Tertium) and Theorem 5 (Theorema Quintum) in "De Revolutionibus Orbium Coelestium" are applications of Ptolemy's theorem to determine respectively "the chord subtending the arc whereby the greater arc exceeds the smaller arc" (ie sin(a-b)) and "when chords are given, the chord subtending the whole arc made up of them" ie sin(a+b). A minor adjustment of the given geometry yields the respective cosines.Neil Parker (talk) 11:30, 28 October 2019 (UTC)[reply]

In a quadrilateral of consecutive sides a,b,c,d and diagonals p,q, the difference (ac+bd)^2-(pq)^2 can be expressed in terms of the cosine of the average of two opposite angles. I wonder if this can be mentioned. 2601:5C4:4280:11F:E847:707E:1655:1DF (talk) 12:59, 17 February 2023 (UTC)[reply]