Talk:Ptolemy's theorem

message: 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)
thread_title:  moved 

message: 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 
:: \overline{P_1P_3}\cdot \overline{P_2P_4}=\overline{P_1P_2}\cdot \overline{P_3P_4}+\overline{P_1P_4}\cdot \overline{P_2P_3}
:Here \overline{P_iP_{i+1}}=\overline{P_{i+1}P_{i}} for  i=1,\ldots,4 denote the lengths of the four sides of the quadrilateral (with indices taken modulo 4), the two diagonals are then \overline{P_1P_3} and \overline{P_2P_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: 
:: \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)
thread_title: ABCD?

message: 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)
replies: text: 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)

text: Most sources treat them together.  Here's a book that treats the inequality as a strengthening of the converse of the theorem:   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)

text: 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)

text: And thanks for fixing my typos, Michael Kinyon. Dicklyon 00:01, 8 October 2006 (UTC)
thread_title: Merge?

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title:  Cleanup tag 

message: 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)
thread_title:  SmackBot's vandalism 

message: 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)
thread_title:  Re-organisation 

message: 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.
: \overline{AC}\cdot \overline{BD}=\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{AD}
I suggest using one of:
: AC \cdot BD = AB \cdot CD + BC \cdot AD
or
: AC \times BD = AB \times CD + BC \times AD
Nturton (talk) 11:32, 7 May 2009 (UTC)
replies: text: 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)

text: |AC|\cdot |BD|=|AB|\cdot |CD|+|BC|\cdot|AD|
Bo Jacoby (talk) 22:10, 27 March 2010 (UTC)
thread_title:  Confusing notation 

message: 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)
thread_title:  Re-organisation 

message: 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)
thread_title:  Proofs 

message: 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)
thread_title:  collinear 4-tuple 

message: What exactly is meant by "Complement of Pentagon Chord"? Angles have complements. Do chords have complements??--WickerGuy (talk) 18:43, 2 March 2012 (UTC)
replies: text: That whole section is rather dubious should be removed or at least rewritten.--RDBury (talk) 03:13, 5 March 2012 (UTC)
replies: text: 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)
thread_title:  Bad terminology 

message: 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)
replies: text: As long as their are readded with proper sourcing there is probably no issue. --Kmhkmh (talk) 17:49, 20 May 2016 (UTC)
replies: text: 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)
replies: text: 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)
thread_title:  Removal of historically important examples 

message: 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)
replies: text: Slightly modified the text by introducing an intermediate \delta. Actually, the sum of the 4 inscribed angles subtended by the 4 sides of the quadrilateral is \pi, not 2\pi. --Roland (talk) 05:14, 25 April 2017 (UTC)
thread_title: Proof by trigonometric identities

message: 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)
replies: text: 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)
replies: text: 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 = eia
B = eib
C = eic
D = eid
We have the identity 
(B−A)(D−C) + (C−B)(D−A) = (C−A)(D−B).
Then (B−A)(D−C) = (iei(a+b)/2 |AB|)(iei(c+d)/2|CD|) = −ei(a+b+c+d)/2|AB||CD|,
and (C−B)(D−A) = (iei(b+c)/2|BC|)(iei(a+d)/2|AD|)= −ei(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)
thread_title:  Proof using complex numbers 

message: 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)
thread_title:  adding modern new proofs for Ptolemys theorem 

message: 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)
replies: text: 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)
thread_title:  Application to cosine (a - b) 

message: 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)
thread_title:  Identity for non-cyclic quadrilaterals.