# Talk:Triangle/Archive 1

## Euclidean Geometry

Someone should probably bring up that this page, while a good discussion, is only as far as I know valid for Euclidean geometry. There is no inherent problem with this, but someone should bring up triangles in other geometries. 68.6.85.167 22:53, 2 June 2006 (UTC)

Speaking of Euclid, didn't he prove that three of the congruence "schemes" could be proven from the fourth? That is, you need only have at most one postulate; the rest can be theorems. (I think his method was to start with SAS and prove the others from there—even without using the nonparallel postulate.) OneWeirdDude (talk) 02:24, 17 August 2008 (UTC)

## Major browser compatibility problems with article

1. The .svg format is not supported by well over 90% of browsers in use. Use animated GIFs instead if animation is absolutely necessary (cannot be replaced by a still image or a series of still images).

3. Code like this: ":$c^2 = a^2 + b^2 \,$" (angle brackets replaced) is producing this:

"Failed to parse (Can't write to or create math output directory): c^2 = a^2 + b^2 \,"

4. Code like this: "Image:Pythagorean.svg|Pythagorean.svg|thumb|The Pythagorean theorem" (double brackets removed) is producing this:

"Error creating thumbnail: Error saving to file /mnt/upload3/wikipedia/en/thumb/d/d2/Pythagorean.svg/180px-Pythagorean.svg.png The Pythagorean theorem"

5. The ":$...$" code in the Using Coordinates section is producing blank sections with a punctuation mark or empty Wiki quote box in it.

I assume you mathematicians have plugins that render all this -- try taking a look on a normal computer.

• Mediawiki (the software that runs wikipedia) renders SVGs and maths formulae as images (.png) which are compatible with almost all browsers. You don't need special plug-ins or anything. There appears to be a serious caching problem with all the images in article, though. I've made a null edit which may help. --Bob Mellish 16:56, 19 April 2006 (UTC)

## Congratulations

Oh, dear! The diagrams in this page are GOOD! Whoever did them did a good job! Pfortuny 21:49, 31 Mar 2004 (UTC)

I second that. Fantastic page with wonderful diagrams. I learned more than I ever expected (or wanted to) about triangles. - Plutor 14:46, 3 May 2004 (UTC)
Kudos to whoever made those diagrams, they make everything MUCH clearer.

I've found the article generally clear, but I have some criticisms. The first one is about the use of term "equal" (instead of "congruent) and the confusion of angles with their amplitudes. For instance:

 In Euclidean geometry, the sum of the angles α + β + γ is
equal to two right angles (180° or π radians). This allows determination
of the third angle of any triangle as soon as two angles are known.


should IMHO be:

 In Euclidean geometry, the sum of the internal angles is a straight angle.
This allows determining the amplitude of the third internal angle of any
triangle once the amplitudes of the others are known.


Similar confusions exist between segments and their lengths.

Am I wrong? Don´t think so. Is it possible for a triangle to have three acute angles?

It's quite possible. An equilateral triangle has three 60° angles. 60 < 90, so they're all acute. - Plutor 16:14, 17 May 2004 (UTC)
Thanks, Plutor, my brain went out for lunch.

I'd like to know where the shape called trochoid fits into the grand scheme of triangles. It's been a while since I've touched geometry, so please forgive me. I don't know if it's the proper term, but it is used to describe the shape of the rotor in the Wankel engine found in the Mazda RX-7/8 and others vehicles. I'd have to say it's a 2D shape with a 1D surface, and basically an equalateral triangle with curved, instead of straight, sides. TimothyPilgrim 13:10, Jun 10, 2004 (UTC)

## Equilateral Triangles - Another Way to Calculate Area

In my geometry class last year, we learned that you could calculate the area of an equilateral triangle.

It is:

((s^2)(square root of 3))/4.

That should be read: Triangle side squared times the square root of 3. That product is then divided by 4.

However, I'm new to Wikipedia editting. I don't know how to create the mathematical symbols to present that formula. I'm also not sure where that fits into the article. If you are able of incorporating this into the article, I would be most grateful. --Acetic Acid 05:10, 23 July 2005 (UTC)

Try ${\displaystyle {{\sqrt {3}} \over 4}s^{2}}$ using ${\sqrt{3} \over 4} s^2$ --Henrygb 10:09, 30 October 2006 (UTC)

"Triangles can not and do not exist in reality, they are purely theoretical mathematical objects. Common misconceptions may regard pyramids as "big triangles," but though they may be triangular, a pyramid is its own geometrical figure.

I don't think the above is particularly useful. If anyone wants to discuss this, we can. Paul August 13:09, August 25, 2005 (UTC)

## ∆

links here, but isn't that the greek letter Delta (letter)? ���� 213.112.14.187 07:54, 8 March 2006 (UTC)

Yes it is, that should probably be fixed. Although who looks for the actual Greek symbols on the English Wikipedia? --Lomacar 00:09, 12 April 2006 (UTC)
Δ redirects to Delta (letter), to Triangle--Henrygb 16:35, 24 July 2006 (UTC)16:34, 24 July 2006 (UTC)
I've changed this to a disambiguation page. There are enough confused users who don't get the distinctions, don't have them on their keyboards or text programs, or who can't visually see the difference on the screen. Redirect is needlessly confusing. --lquilter 18:10, 4 February 2007 (UTC)

## Equilateral vs Equiangular

"An equilateral triangle is NOT equiangular, i.e. all its internal angles are not equal—namely, 69°"


Is this trying to say that an equilateral triangle is not merely equiangular, or in other words doesn't simply have 3 equal arbitrary angles, the angles must be 60° but the defining characteristic is the 3 equal sides? Because if that is the case it is terribly written. Regardless, it had me severely confused. --Lomacar 00:05, 12 April 2006 (UTC)

No, it was just an act of vandalism; now corrected. Thanks for the warning. -- Jitse Niesen (talk) 02:49, 12 April 2006 (UTC)
Wow, vandalising the triangle article, you know you are cool when...--Lomacar 07:56, 13 April 2006 (UTC)

## Wrong formula for the area of the triangle

The formula

${\displaystyle S={\frac {1}{2}}{\sqrt {{\begin{vmatrix}x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}z_{1}&z_{2}&z_{3}\\x_{1}&x_{2}&x_{3}\\1&1&1\end{vmatrix}}^{2}}}}$

was wrong. A counter example is x=(1,0,1), y=(0,1,1), z=(0,0,1). (Actual result: 1/2, result of formula: sqrt(3)/2)

I replaced it by

${\displaystyle S={\frac {1}{2}}\left({\begin{vmatrix}x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\1&1&1\end{vmatrix}}+{\begin{vmatrix}y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\\1&1&1\end{vmatrix}}+{\begin{vmatrix}z_{1}&z_{2}&z_{3}\\x_{1}&x_{2}&x_{3}\\1&1&1\end{vmatrix}}\right)}$

A proof for this can be found at http://mcraefamily.com/MathHelp/GeometryTriangleAreaVector2.htm -- anonymous

When I try out the first formula, I get the correct answer:
${\displaystyle S={\frac {1}{2}}{\sqrt {{\begin{vmatrix}1&0&0\\0&1&1\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}0&1&1\\1&1&1\\1&1&1\end{vmatrix}}^{2}+{\begin{vmatrix}1&1&1\\1&0&0\\1&1&1\end{vmatrix}}^{2}}}={\frac {1}{2}}{\sqrt {1^{2}+0^{2}+0^{2}}}={\frac {1}{2}}.}$
Perhaps you were confused by the names of the variables? The variable x2 is not the second coordinate of the point x, but the x-coordinate of the second point. I renamed the variables in an attempt to clearify.
I'm not sure that the formula you replaced it by is correct; try it out with (1,0,1) and (1,1,0) and (0,1,1). In terms of the cross product, the second formula you give is
${\displaystyle S={\frac {1}{2}}{\big (}|u\times v|+|v\times w|+|w\times u|{\big )},}$
which is not the same as
${\displaystyle S={\frac {1}{2}}{\big |}u\times v+v\times w+w\times u{\big |}.}$
-- Jitse Niesen (talk) 09:30, 30 June 2006 (UTC)

## Proof that angles in a triangle sum to 180 degrees

Regarding the proof at http://www.apronus.com/geometry/triangle.htm : Of course it assumes the parallel postulate, but that doesn't make it wrong. Every proof assumes certain axioms. -- Jitse Niesen (talk) 06:26, 1 November 2006 (UTC)

Actually, it goes furthe that. If you have a geometry without the parallel postulate (such as spherical or hyperbolic geometry), then the angles of a triangle don't sum to 180 degrees.

Because it only assumes the parallel postulate it is merely a restatement of it. Had it assumed other axioms it would qualify as a proof. Since it does not, it is not a proof, merely a restatement.

That does not matter. A proof that uses only one axiom is still a proof; do you have a source that claims otherwise?
As an aside, it's not clear to me that it only assumes the parallel postulate. Which version of the parallel postulate are you thinking of, and how would you proof <)BAC = <)B'CA? -- Jitse Niesen (talk) 00:48, 22 November 2006 (UTC)

## Monster formula

This is cute, but excessive!

${\displaystyle S={\frac {1}{2}}{\sqrt {\left(\det {\begin{pmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}y_{A}&y_{B}&y_{C}\\z_{A}&z_{B}&z_{C}\\1&1&1\end{pmatrix}}\right)^{2}+\left(\det {\begin{pmatrix}z_{A}&z_{B}&z_{C}\\x_{A}&x_{B}&x_{C}\\1&1&1\end{pmatrix}}\right)^{2}}}.}$

Why not just use difference vectors and a cross product: A=(Ax,Ay,Az), B=(Bx,By,Bz), C=(Cx,Cy,Cz)

Area=1/2*abs((B-A)x(C-A)) = 1/2*abs(B-A)*abs(C-A)*sin(angle).

Tom Ruen 03:24, 22 November 2006 (UTC)

## Equal Triangles ?

Please add a section dealing with equal triangles. The Ubik 18:08, 4 December 2006 (UTC)

## What should be included

Once again, I am reverting this entry to the way it was when I put in 6 extra formulae for the area of a triangle all based on 0.5absinC. The Wikipedia articles should provide a source of reference for everyone and should be as complete as possible. A lot of my own students use this to check basic formula and these entries of mine are necessary. The first set of three formulae are well known but the second set of three are not so well known and help reiterate the symmetry of the sine curve. One man's trivia is another man's reference. Dont take it upon yourself to police this page. Be true to the Wikipedia ideal - a comprehensive source of reference. Sorry that my IP address keeps changing. Not my fault. 81.158.253.8 23:51, 23 January 2007 (UTC)

Reply by Oleg Alexandrov (talk) below:

(a) Here is the Relevant diff.

(b) You can create an account as requested, which would make discussion more productive.

(c) I would argue that the text

 If one uses ${\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}}$ and ${\displaystyle \sin C={\sqrt {1-\cos ^{2}C}}}$ and also the formula shown above, then one arrives at the following formula for area ${\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}$ [Note that, this is a multiplied out form of Heron's formula] Using a symmetry argument these three formulae also give the area (compare above S = ½ab sin γ.) ${\displaystyle {\frac {1}{2}}ab\sin C={\frac {1}{2}}bc\sin A={\frac {1}{2}}ca\sin B}$ Using the property of the sine curve, namely sin X = sin (180-X) one arrives at three more formulae ${\displaystyle {\frac {1}{2}}ab\sin(A+B)={\frac {1}{2}}bc\sin(B+C)={\frac {1}{2}}ca\sin(C+A)}$

is not necessary because

1. The derivation of Heron's formula belongs, if anywhere, at Heron's formula article. Heron's formula itself is mentioned already in the article, one section below this text.
2. The formulas
${\displaystyle {\frac {1}{2}}ab\sin C={\frac {1}{2}}bc\sin A={\frac {1}{2}}ca\sin B}$
and
${\displaystyle {\frac {1}{2}}ab\sin(A+B)={\frac {1}{2}}bc\sin(B+C)={\frac {1}{2}}ca\sin(C+A)}$
are trivial deductions (yes, even for high school students) from the formula
S = ½ab sin γ
already in the article. Oleg Alexandrov (talk) 03:14, 24 January 2007 (UTC)
Oleg is correct, if understated.
1. I concur that we do not need to extend this article with an out-of-place "proof" of Heron's formula when it has its own section and its own article.
2. Giving three different forms, which merely depend on an irrelevant free choice, is bad mathematics and bad writing. This is not a right triangle, where one angle (one side) is special. Sorry, anon, but your students must learn to fit a reference formula to a specific circumstance. That applies, not just here, but everywhere.
Monitoring edits for quality control is a shared responsibility. Your edits were deleterious, in Oleg's view and in mine, so we have improved Wikipedia as a resource by reverting. This is not a personal reflection on you, and we look forward to many fine contributions to your credit should you choose to establish an account. (Accounts are a Good Thing. They give greater privacy by suppressing your IP address, they allow you to edit and talk with a consistent identity, and they provide a reliable page where others can contact you.) If you do intend to edit mathematics articles, we invite you to join our discussions at Wikipedia:WikiProject Mathematics, and to refer to our Manual of Style for mathematics, to our mathematics conventions, and to our citation guide. --KSmrqT 04:07, 24 January 2007 (UTC)
I agree with Oleg and KSmrq. I have again restored the article to its original version. Paul August 04:49, 24 January 2007 (UTC)

I disagree with the statement that these are "trivial deductions ... even for high school students". Actually, I teach in the UK not the USA. It's quite possible, but unlikely, that every high school student might already know 0.5absinC = 0.5bcsinA = 0.5casin B but I suspect that few of them know and fewer could explain that 0.5absinC = 0.5absin(A+B). It's nice for you that you have taken it upon yourself to police this article but it's very irritating for me who would like to see it as a comprehensive reference guide. So once again, I am reverting the article. Thank you

Since my last note, I have moved things around a bit so that it flows better and in particular a link is made connecting 0.5absinC = 0.5bcsinA = 0.5casin B with the sine rule. This adds weight to the necessity of keeping these formulae. Thank you

I don't see what UK v USA has to do with anything. They are trivial deductions. Noone said that they "know" that the three expressions are equal to each other, but it does follow immediately from the area formula. In fact, they are the same formulae, and pretending that they are different doesn't help anyone. The change from ${\displaystyle \sin A}$ to ${\displaystyle \sin(A+B)}$ is slightly less trivial, but still straightforward and not really helpful to an article on triangles. At most, it justifies adding one more formula, not three. This article should not be "comprehensive" in a way that duplicates information which really only needs to be at Heron's formula. Giving a proof here, mentioning it in two separate sections, causes unnecessary confusion. JPD (talk) 13:01, 24 January 2007 (UTC)

Disagree. The article should be comprehensive. All this talk of what is "trivial" suggests that you want the article to be written for mathematicians whereas I want it to be written for the masses. The mathematicians probably already know all the formulae so they wont even want to visit this entry on Wikipedia. The entry has to stand as I last edited it on the basis that it is good reference material for the masses. What I am doing *IS* helpful and what you are doing *IS NOT* so kindly stop deleting my work. Thank you

Firstly, I haven't deleted your work, so I don't know what you're talking about. Secondly, being comprehensive should not be an excuse for saying the same thing more than once, making references to things that are only mentioned later or anything like that. The Heron's formula info is just a mess. Thirdly, as you have written it, it is bad reference material for the masses, because it suggests that Sab sin C and Sbc sin A[/itex] are actually different formula from Sab sin γ. And that's on top of the fact that the article quite clearly says that in this article the angles of the triangle are α, β and γ. Where did A, B and C come from? The addition does not make the article more comprehensive, it just adds, as KSmrq says, bad mathematics and bad writing. JPD (talk) 15:08, 24 January 2007 (UTC)

Somebody keeps deleting my work - not you maybe. If anything is trivial, it is that γ is the same as C. However, I have taken this on board and added explanatory text. Now leave it alone unless you want to enhance it but not by deleting my work.

No, unless you are used to labelling angles with the name of the vertex, the C, and the A and B, come from nowhere. In contrast, the use of α, β and γ is explained even to those who may not be used to it. Why suddenly change notation in the middle of an article? Simply inserting formulae in the format you teach them does not make the article more comprehensive, just more confusing. Even if it is helpful to mention the symmetry, describing your formulae as another three formulae is plain unhelpful. As fro much of the other material, it is worth remembering that one of the ways in which Wikipedia can best be comprehensive is through links to other articles, meaning no one article has to contain everything vaguely related to the topic, and that Wikipedia is a collaborative effort. Your work is not only your work, it is also either enhancing or disrupting other's work. JPD (talk) 18:28, 24 January 2007 (UTC)

This discussion is interesting but I have to say that I agree with the anonymous poster. The extra formula are useful and I am fascinated to find so many of you (JPD, KSmrq, Oleg) kicking up such a fuss. Let's leave the poor guy (girl?) alone. Troy Prey 19:29, 24 January 2007 (UTC)

I have added a note, and a value for the diameter of the circumcircle, which may make some of the points the anon wants, without so much verbiage. I hope this will assist convergence to consensus. Septentrionalis PMAnderson 19:53, 24 January 2007 (UTC)

Thanks but no thanks. I have left your amendment but also included the original article as I last left it. At least I was respectful enough to do that. If my points are so trivial then what is so special about things like 30-60-90 and 45-45-90 triangles? Arenot they trivial in the light of the whole article? Please think about it and remember this has to be a comprehensive reference article for all. Sorry but I refuse to give in on this one. And BTW, I am male.—Preceding unsigned comment added by 81.158.253.8 (talk) 20:11, January 24, 2007

Since I agree with the comments of Oleg, KSmrq, JPD and Septentrionalis above, and I think that Septentrionalis' version better than your's, I have restored his version. Please understand that editing on Wikipedia is a collaborative process. No single editor can impose their views on the article. Please read WP:CON, and WP:3RR. Paul August 20:45, 24 January 2007 (UTC)

I am beginning to suspect that you are all the same person but never mind that. Yes, I agree it should be a collaborative process but does that mean democratic? Shall I simply go and find more people than you can find who agree with me? Do you feel that that is the way forward? I dont! I am a Mathematician and a Maths teacher and I understand that Wikipedia is trying to be a *comprehensive* source of reference and that is what I am trying to achieve here. These formulae are useful so please get off your high horse and leave them alone. I am finding this a little irritating.

No, we are not all the same person; we simply agree. Please read WP:Consensus and WP:3RR before you revert again. Septentrionalis PMAnderson 21:52, 24 January 2007 (UTC)

Ok, I have created an account now and had a look at the link. My opinion is "more is better than less". It's better to have a page with more information rather than less even if it helps just one person. But realistically, leaving all the formulae in will help a *LOT* of people and that is what Wikipedia is all about. Look up the meaning of encyclopaedia. In my dictionary it says "... dealing with the whole range of human knowledge..." What you are proposing is to have less information which does not make sense. There is an elitism going here amongst some of you saying that like "trivial deductions ... even for high school students". Perhaps where we differ is that you feel that this is a source for high level Mathematicians whereas I believe that this is a source for all especially school children. Anonymath 22:02, 24 January 2007 (UTC)

Thanks for creating an account. What do you think of the present version of the article? Paul August 22:41, 24 January 2007 (UTC)

Yeah, it's better. I could live with this for now but why delete the proof of Heron's formula from trigonometric considerations? Anonymath 23:02, 24 January 2007 (UTC)

It wasn't a proof; it was an exercise in hand-waving. Most people who can convert that into a proof don't need a proof at all; those who can't, won't benefit. So the chief effect was to include Heron's formula twice. Septentrionalis PMAnderson 23:36, 24 January 2007 (UTC)

P.S. I think it might be better to label the angels A, B and C and not alpha, beta and gamma but I havenot attempted to do this myself. Anonymath 23:14, 24 January 2007 (UTC)

We've already labelled the points A, B, and C. Using the same letters for the magnitudes of the angles is just asking for confusion. If we were doing spherical trig, where the symmetry between sides and angles is real, it might be worth the cost. Septentrionalis PMAnderson 23:36, 24 January 2007 (UTC)
Thank you for creating an account; amusing choice of ID. :-)
I will first address the issue of who you are speaking with. The participants in this discussion (so far) are Oleg Alexandrov, myself (KSmrq), Paul August, JPD, Troy Prey, Septentrionalis, and you (now Anonymath). I am not at all familiar with Troy Prey, and only a little familiar with JPD; but Oleg, Paul, Septentrionalis, and myself are frequent contributors to mathematical discussions and articles. Paul has the distinction of being recently chosen by Jimmy Wales to be on the Wikipedia Arbitration Committee, based on 220 support votes and 18 oppose votes (92.44% in favor), and most of us have been active within Wikipedia for quite some time. So I can assure you that we are distinct persons in real life, and that you have an educated, experienced, and fair group to talk with.
Wikipedia operates almost entirely by consensus, which some have likened to mob rule, and others to populist democracy. Sometimes that means that a group of wise voices prevails, sometimes the opposite. In issues of fact, our definitive arbiter is a reliable published source, such as a peer-reviewed journal article. That is not at issue here. In matters of what should be included or not, opinions vary widely. Some support having a detailed article on anything that anyone might want. The mathematics community tends to be tolerant, but somewhat more conservative. The important thing for you to understand as we proceed is that you cannot dictate, and any attempt to do so will harm your cause as you try to sway a consensus your way.
I must especially caution you not to constantly revert against consensus. Wikipedia views that as a serious disruption, and may block your editing privilege (including your IP addresses) if you persist. But I hope we can talk this out.
You will not get far with me by charging elitism. I urge you to read some of the many answers I have posted on the mathematics reference desk to see how much I try to speak to a very broad audience. Also, I have experience teaching (as do others in this conversation), and your arguments about what students need have not persuaded me.
Perhaps we can reach a mutual accommodation, as Septentrionalis has tried to do. Perhaps you will never be completely satisfied with the outcome. I do urge you to try, and to adapt to this peculiar thing called Wikipedia. We know that you must climb a steep learning curve as you integrate into the community, and we will try to be as friendly and helpful as we can. And we do sincerely appreciate your desire to contribute positively, and the efforts you are making to do so. --KSmrqT 23:43, 24 January 2007 (UTC)

(1) Firstly, thanks for the welcome. (2) Regarding the use of A, B and C to mean both the angle and the name of the vertex - this is universal practice in all the UK textbooks. Perhaps the universal practice in US textbooks is to use alpha, beta and gamma for the angles so maybe this is a UK vs USA thing after all. I personally think it's better and simpler and less confusing to use A, B and C. In fact, it adds to the confusion to use alpha, beta and gamma especially gamma because hardly any younger (UK) students have even heard of it let alone seen what it looks like. (3) I was almost happy the way it was left yesterday and said so but I am unhappy with the change from 180 to pi - this is what I was saying about elitism yesterday. How accessible is this if you talk in radians instead of degrees? Either talk in both or just in degrees but dont talk just in radians. (4) I have been exploring all the "rules" and "guidelines" about Wikipedia and how it works and note that it is not intended to be a democratic process. (5) To KSmrq: Which age and level students do you teach? I teach a broad age group - everything from 10 to 18. I am surprised at your suggestion that students dont need what I am suggesting. We dont teach radians until they get to 17 and even then it's only for those who have chosen not to drop Maths at 16. (6) Keep it triangle  :-) Anonymath 11:03, 25 January 2007 (UTC)

I would be surprised to find any practice universal, even within a single country. I would have no strong objection to revising the notation root and branch; I do strongly object to being inconsistent about whatever the notation is. I still think it unhelpful to use A in two senses; but if other editors think it worthwhile, fine. I won't fight to keep π; but I certainly knew what a radian was when I was 17; and the article does define it. Septentrionalis PMAnderson 22:02, 25 January 2007 (UTC)

## Simplest Area Formula

Why don't I see the simple formula A= 1/2 bh prominantly at the top of the section on "area of a triangle"?--Lbeaumont 01:24, 30 March 2007 (UTC)

I was wondering the same thing. --Yath 08:55, 31 March 2007 (UTC)

Haha me too. I mean, there are people (like myself) out there that don't know what QxT/(Z+A)-%^\$##%@#% is. In my opinion, all math articles should have a simpler explanation (execpt for calculus, trigonometry, and so on where there is no simple explanation). Abcw12 06:26, 5 June 2007 (UTC)

Ok, I added that formula in the beginning paragraph. However, I don't have any experience with the wiki "math" block, so I put it in text only. ROBO 04:21, 6 October 2007 (UTC)

## right triangle

Can someone write an article at right triangle so that it isn't just a redirect? As right triangles are so important in life, carpentry, trig, etc. 70.55.84.34 08:48, 5 October 2007 (UTC)

I certainly think a section on right triangles would be good. Whether it would get big enough to warrant having its own page, I'm not so sure. -- Steelpillow 20:45, 5 October 2007 (UTC)
Having said that, I just followed a link in the page to Special right triangles. Is this anything like what you had in mind? -- Steelpillow 20:49, 5 October 2007 (UTC)
Actually, I was thinking of a more generalized right triangle article. 70.55.84.154 05:17, 8 October 2007 (UTC)

## solving a triangle

I looked in vain for methods of finding the remaining attributes of a triangle when only some are known. For example, when two sides & the included angle (say a, c and B) are known, it can readily be seen that tan C = (c*sin B) / (a - c*cos B). Then the sine rule yields the other side, b. I'd be happy to add this, with a proof, and a statement of the sine rule itself. But if one of you activists would like to add it to suit an existing style, please go ahead; I'll wait for a week or two then add it & hope for the best. John Wheater 10:18, 7 November 2007 (UTC)

## Centroid/barycenter confusion

"The centroid (yellow), orthocenter (blue), circumcenter (green) and barycenter of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line)."

I thought the centroid and barycenter were the same thing? Or is this talking about the centroid of the triangle and the barycenter of the nine-point circle? I apologize if I'm way off-base here, -- I'm no expert here, and this seems confusing.

Thanks. CSWarren (talk) 17:28, 18 November 2007 (UTC)

## Congruence

Way back in 2004 someone queried the use of the term congruent for angles. No one responded. It is to be assumed, then, that editors agree with the objection. In fact, in standard usage congruence never applies to angles or sides, only to figures. This is all very straightforward and clear, since the standard term equal works perfectly well for angles and sides that are simply measured numerically. See Congruence (geometry); see also major British and American dictionaries: SOED, and M-W Collegiate (Congruent "2: superposable so as to be coincident throughout"; there is no "throughout" for sides or angles, since they are not compound as geometrical figures are).

– Noetica♬♩Talk 21:06, 10 December 2007 (UTC)

I agree that, in basic articles, we should use the simplest possible precise language. Therefore we should use "equal in length" for sides called "isos" (equal) by Euclid. Use of the word "congruent" for sides and angles just muddies the waters (especially when we are explaining congruence of triangles) without improving the meaning. Dbfirs 08:36, 13 February 2009 (UTC)
Good, Dbfirs. Of course, we know that there are more rigorous formalisations of all these things, including some that apply a notion of congruence to angles and line segments. But for at least two reasons this is unhelpful in articles such as this one:
1. It would have to be done uniformly, rather than employing some Hilbertian definitions and terminology while retaining many that are traditional, and more or less Euclidean. For example, the "angles" that are spoken of as congruent are not strictly the same entities as the "Euclidean" angles of common usage.
2. As you suggest, this is one of a suite of basic articles, and while rigour is of great importance in them (and I myself have edited to enhance that rigour), clarity is paramount.
¡ɐɔıʇǝoNoetica!T– 09:10, 13 February 2009 (UTC)
Yes, this is essentially an article on Euclidean triangles, with just a mention of generalisation at the end, so it seems sensible to preserve Euclid's usage in this and basic similar articles on Euclidean geometry (except where such usage has clearly changed over the intervening 2000 years). Thanks for your support. I wonder if anyone who disagrees could put a rational counter-view? Dbfirs 11:12, 13 February 2009 (UTC)

## Error in a definition

Under the section "Basic Facts" of this article there is an error in defining a word. In the third paragraph where it explains the definition of a triangle, the definition states an assumption about angles and then uses the word itself (angle) in its own defintion. This is the text as it appears:

The reason it has the name "triangle" is because its a compound word with words about the triangle. Meanings: Tri-Angle: Tri-The word for the number 3, like 1 is uni, 2 is bi and etceteria. Angle: Probably everyone knows this word, it means a diagonal line of any angle.

The definition of an angle is the union of two nonopposite rays emmanating from a commom point.

76.84.115.44 (talk) 06:03, 16 January 2008 (UTC)Andy Ransone

Thanks for pointing this out. I've removed that strange paragraph. You could have done the same: anybody can edit this encyclopedia. The etymology of the word triangle can be found here.  --Lambiam 16:57, 16 January 2008 (UTC)

## Isn't there another formula?

A=.5ab*sine of included angle —Preceding unsigned comment added by 66.65.139.242 (talk) 07:52, 19 January 2008 (UTC)

This is the first formula given in the section Using trigonometry.  --Lambiam 17:10, 19 January 2008 (UTC)

## Graphing a Triangle on a Cartesian coordinate system

What equation is there for graphing a triangle? You can use inequalites separated by AND. Thx. —Preceding unsigned comment added by KyuubiSeal (talkcontribs) 14:33, 16 April 2008 (UTC)

## Another Triangle Formula

This page should mention that, in the case of a right triangle, the multiplication of the catheti is equal to the multiplication of the triangle's height (perpendicular from the hypothenus to the right angle) and the hypothenus. —Preceding unsigned comment added by 216.113.19.14 (talk) 23:45, 9 June 2008 (UTC)

## Merge with Polygons article?

There is a note at the bottom of the page saying "It has been suggested that this article or section be merged with Polygons. (Discuss)". As stated at the beginning of the article "A triangle is one of the basic shapes of geometry". It has a large enough set of properties, definitions, equations and concepts pertaining to it specifically to merit being assigned its own article (just look at current size of the article). Keeping Triangle as its own page makes information specific to triangles much easier to find and deliniate than if included in a larger article on Polygons. These are of course, my personal opinions (Although the talk-page guidelines seem to frown upon stating opinions in general, the reference to the suggested merge seems to invite opinions on the matter and links to this discussion page). GameCoder (talk) 00:18, 16 June 2008 (UTC)

Hi GameCoder, don't worry, in this case you're expressing an opinion about how an article should be written, which is exactly what talk pages are for. I agree with you here and I'm not sure why a merge was suggested. Whoever suggested it doesn't seem to have given any rationale, so I'll probably remove the tag. Gimme danger (talk) 02:33, 16 June 2008 (UTC)

## Inverse Functions and Example

The subsection Inverse Functions now has a specific example regarding HOWTO solve a problem involving the inverse Trig Functions. While it might be required, I do not believe this is appropriate for an encyclopedic article. Does everyone here agree/or disagree with my opinion. Furthermore, I do believe that this section itself requires a little bit (not a lot, just a little) cleanup and addition of all the arc functions. Aly89 (talk) 15:24, 20 October 2008 (UTC)

I agree that this is inappropriate, and that the section is incomplete and needs cleanup. Please do so if you have the time. --Gimme danger (talk) 16:01, 20 October 2008 (UTC)
Done. Any criticism on my cleanup, or on any other sections of the page would be appreciated. Aly89 (talk) 18:05, 9 November 2008 (UTC)
- I went ahead and added a short nod to sin−1 notation at the bottom of the simple inverse trigonometric definitions. I think people unfamiliar with the simple definition of an inverse trig function run a reasonable risk of being unfamiliar with arc notation (although I can certainly see this as superfluous information also). Pinochet (3) (talk) 05:46, 23 June 2009 (UTC)

## Equilateral triangle existence

My son says that there cannot be a true equilateral triangle in reality, only mathematic theory, because it's existence would cause the destruction of the world. Does anyone out there agree with his theory??

Yes, I do. I have actually attempted to create a true equalateral triangle. I was near success when I suddenly fainted and had a vision that the equilateral triangle (calling itself "Equatrango the Machine") was destroying every other shape known in existence,except triangles. Thus it destroyed our world, which is a sphere. I immediatly discontinued my project when I awoke from this horrible prophecy, and now I only like circles.

Please sign your posts. Anyway I don't really care much whether the angles are 60 degrees or 60.000001 degrees. If it looks equilateral, if the protractor says the angles are about 60, isn't that good enough? --116.14.26.124 (talk) 01:15, 30 June 2009 (UTC)

Your son is, of course, correct in that it is impossible to draw an absolutely perfect equilateral triangle (or any other exact triangle), but destruction of the world requires more drastic measures. Dbfirs 07:11, 30 June 2009 (UTC)

## "Using Coordinates" Section

That formula given to general vertices is wrong. Somebody calculated the determinant the wrong way.

I have no time to fix this right now, but the right formula (I've just recalculated in Mathematica and got this):

1/2 | -(-xb + xc) (-ya + yb) + (-xa + xb) (-yb + yc) | —Preceding unsigned comment added by 201.11.229.230 (talk) 08:08, 18 April 2009 (UTC)

Thanks for pointing out a possible error. I don't use Mathematica, but multiplying out longhand shows that your formula is correct, and also that the article is correct, and simpler. Your Mathematica version has some unneccessary minuses, but is actually the same formula. It just looks different. Dbfirs 06:45, 30 June 2009 (UTC)

## Talk

Does anyone know what this means? "Also, the exterior angles (3 total) of a triangle measure up to 360 degrees." It does not make sense to me. Tom Hubbard 22:21, 3 July 2007 (UTC)

OK, this seems to be my misunderstanding of what is meant by an exterior angle. The article about [[1]] says that an exterior angle is formed by the exterior of the shape. So I was thinking that for an example of an equilateral triangle, the exterior angles would each be 300 degrees. Actually, the exterior angle is found by extending a side of the shape and then measuring the angle. So actually and equilateral triangle has 3 exterior angles of 120 degrees. Sorry for any confusion -- probably the angle article should be more clear. Tom Hubbard 13:16, 13 July 2007 (UTC)

I just corrected the Types of angle -- Steelpillow 10:08, 14 July 2007 (UTC)

I wonder if there's another formula to add for the area of the triangle, based upon dot products of vectors. When you take the vector from point 1 to point 3 as U, and the vector from point 3 to point 2 as V: A = 0.5 * sqrt ((U*U)(V*V)-(U*V)(U*V)). I just derived that based upon the geometric version A=0.5(base)(height), calculating the point of intersection of the altitude along the base, to be V*U/U*U. If this appears right to others, then someone might add it.

Could someone redraw the scalene triangle, It isn't scalene. Ooops - yes it is. It isn't acute, but then it doesn't say it is trying to be - sorry.

Am I the only one who thinks that the geometrical triangle is entitled to reside at triangle? It's far and away the most common usage of the word, and links in the future are naturally going to be made to triangle instead of triangle (geometry). Triangle should have a simple disambig block at the top for the few other meanings. "Triangle" isn't like Orange, which has many possible meanings; it's more like Pentagon, which has a primary meaning and a few derivatives. --Minesweeper 10:03, Mar 6, 2004 (UTC)

I totally agree. I'll have to hear a very good opinion on the current setup in the next few days, or else I'll revert. — Sverdrup (talk) 14:04, 6 Mar 2004 (UTC)

I'd always been taught to use the term right angled triangles - is the usage right triangle a different regional variant? Is mine the regional variant (UK/Ireland)? What does the wider community say? --Paul

In the United States, "right triangle" is the only term I've heard. I don't think I've heard "right angled triangle" before.63.190.97.177 07:51, 14 Mar 2005 (UTC)
UK schools always say "right-angled triangle", but US usage prevails on Wikipedia. I'm not sure about Commonwealth and other English-speaking countries. Dbfirs 09:45, 15 February 2009 (UTC)
The term "right triangle" appears to be a more consistent nomenclature since other angle referenced triangles are called "acute triangles" and "obtuse triangles". I would agree that "right-angled triangles" would be more appropriate if we referred to the others as "acute-angled triangles" or "obtuse-angled triangles".  JackOL31 (talk) 23:42, 26 July 2009 (UTC)
Actually, UK schools do use the terms "acute-angled triangles" and "obtuse-angled triangles", so we are consistent. Dbfirs 08:08, 27 July 2009 (UTC)
I guess that was a tad vague. I was actually referring to this page, not the UK. The UK "right" terms are used twice under "Computing the Sides and Angles" and once under "Inverse Functions". I just now discovered that under SSA, both acute- and obtuse-angled terms are inconsistently used (once each). JackOL31 (talk) 19:30, 16 August 2009 (UTC)

## Sine ambiguity

Because of the unqiue way of Sine function to be positive in both the first and second quadrants, there is a concept known as sine ambiguity which is specifically referred to when solving for angles using the sine rule (arccsin to calculate the angle), or when using the inverse trigonometric function itself. My attempts to find an equivalent reference in the Sine article itself were unsuccessful. Given the sine rule in this article, or the section of the inverse trigonometric functions section, it would be best if the respective sections contained a reference to this or equivalent theorm. If someone knows, where I can find this on any of the articles, please let me know, so I can link to it from this article. Aly89 (talk) 18:12, 9 November 2008 (UTC)

The ambiguity is explained in the Law_of_sines article. All of the inverse trig functions have ambiguities when extended beyond their principal values, of course, but only the sine one affects calculations in triangles. Dbfirs 07:32, 2 July 2009 (UTC)

## Sum of the angles of TRIangle

Sum of the angles is EXACT 3, nothing less and/or nothing more. Someones use 180 or 200 for the value of the sum but three (3) is not divisible (or multiple) by 2 if one wants to be exact. -Santa Claus

TRIangle means 3 sides or vertices, not the interior angles equalling 3 degrees! The sum of the interoir angles is 180 degrees, but there are 3 angles in a triangle since there are 3 vertecies. You must be confused. Either that, or someone in the article left out a word or 2. Abcw12 06:20, 5 June 2007 (UTC)

The sum is 180 degrees; the count is 3. Dbfirs 07:33, 6 September 2009 (UTC)
Correctly stated: The sum of the measures of the three angles equals 180 degrees. It is written correctly in the "Basic Facts" section, however it is written poorly in the "Non-planar triangles" section. One usually assumes "of the measures" with the phrase "The sum of the angles...", but it is technically not correct. It would be considered informal usage. JackOL31 (talk) 20:59, 3 October 2009 (UTC)
... but it is standard usage in British schools, and has been (elsewhere) for 2000 years. Dbfirs 19:10, 29 October 2009 (UTC) Dbfirs 19:10, 29 October 2009 (UTC)
... but it is not standard usage in US schools and elsewhere. Since we are addressing more than one audience, the complete term "sum of the measures", understood by all, should be the preferred usage. Regarding the "2000 years" reference, one must not ignore the fact that the terminology from 2000 years ago is taken out of context from how it is used by many today. Euclid referred to quantities of measure as "magnitudes". These magnitudes could be compared as equal or unequal and could be added or subtracted without the use of real (as in bold, script R) numbers to measure their size. Geometry today is not universally taught within this non-numerical context. Considering the confusion/objection of others regarding this usage, the complete and unambiguous usage that includes "measure" appears to be the best choice for use here. JackOL31 (talk) 03:41, 2 November 2009 (UTC)
That sounds fair. I was brought up with Euclid's usages and implicitly assume magnitudes independent of measurement using real numbers. Dbfirs 17:23, 23 November 2009 (UTC)

For the sentence on the sum of the measures of the angles being 180, one thing unnoticed by most people is that this incorrectly assumes a straight angle's measurement is 0. Technically, a straight angle is 180 degrees, and can be found anywhere on a triangle that isn't at one of its corners. I added "non-straight" to the sentence in this article, but someone reverted me. Any discussion?? Georgia guy (talk) 15:37, 27 February 2008 (UTC)

If it has a straight angle, it has an extra vertex, at that angle. Then it has four vertices, and it is not a triangle anymore, but a degenerate rectangle. Oleg Alexandrov (talk) 15:40, 27 February 2008 (UTC)
Well, we can define a triangle as a polygon with 3 non-straight angles, which is what it technically is, of course. Georgia guy (talk) 15:42, 27 February 2008 (UTC)
There is also the possibility of a degenerate triangle with two angles of 0° and a third of 180°. However, I think no interior angle of a polygon can be 180°, since there is no corner there. But perhaps we should add, for all clarity, that the three corners of a triangle must be non-collinear. "Non-collinear" is already mentioned, but as phrased it is not clear this is a requirement for trianglehood.  --Lambiam 21:44, 27 February 2008 (UTC)
The sum of angles is 180 even if the triangle is degenerate, which was the primary concern here. As such, I think the exposition is already reasonably clear. Oleg Alexandrov (talk) 05:20, 28 February 2008 (UTC)
We could avoid the confusion if schools in the USA would stop teaching the self-contradictory term "straight angle". In the UK it is just a straight line, not an angle at all, though if you mark a point along its length, you can measure an angle of 180 degrees between the line-segments each side of the point. (Ignore this comment, I'm just having a rant!) Dbfirs 08:28, 9 April 2009 (UTC)
An angle is defined as a figure formed from two rays sharing a common vertex. As observed from this definition, the term "straight angle" is neither contradictory nor self-contradictory. There is a distinct, although subtle, difference between a line (points in straight alignment) and a straight angle (an angle whose measure is 180 degrees). While a straight angle forms a line, a line does not form a straight angle. For a line, there is no given angular rotation about a vertex, per se. When referring to angles, one should maintain terminology consistent with angular rotation, such as acute, obtuse, straight, reflex and full. It is no more correct to substitute "line" for "straight angle" than it is to substitute "circle" (all points equidistant from a center point) for "full angle" (an angle whose measure is 360 degrees). JackOL31 (talk) 20:16, 5 September 2009 (UTC)
... and this disputed definition leads to much confusion! I prefer the definition on this side of the pond, so that readers do not imagine angles where (as you correctly say) there aren't any. Dbfirs 07:24, 6 September 2009 (UTC)

I agree, one shouldn't imagine angles where they aren't any (the original posters confusion). However, straight angle is the appropriate term when there is an angle rotation of 180 degrees, meaning an angle exists. There is no contradiction with that usage, your preference notwithstanding. However, substituting "line" for "straight angle" when working with angular rotation is what leads to the confusion. Readers shouldn't imagine no angle (line) where there is one (straight angle). As mentioned earlier, it's the same mathematical error as substituting circle (points in a curved line) when full angle (rotation of 360 degrees) is warranted. One should not substitute lines when angle rotation is required. We could avoid the confusion if schools in the UK would stop teaching the term "line" incorrectly. JackOL31 (talk) 04:17, 7 September 2009 (UTC)

We'll just have to agree to differ about the language used. We don't teach the term "line" incorrectly. We use "line" when, as you point out, there is no angle. I suppose the British expression "angle on a straight line" does sound a bit clumsy. Dbfirs 23:39, 11 October 2009 (UTC)
I have no issue with you using the term line, although I find it befuddling. How one can use a term that according to you implies no angle for the concept of an angle whose measure is 180° is beyond my understanding. However, the issue I do have is your statement that the use of the term "straight angle" is contradictory and therefore incorrect. In my previous post I've cited the definition which illustrates that no contradiction exists between the term and the definition. I guess I am not sure whether you are now saying that the term "straight angle" is an acceptable term to use to convey the meaning of "an angle" whose measure is 180°? If not, could you please explain your mathematical (geometrical) issue with the term? JackOL31 (talk) 18:19, 17 October 2009 (UTC)
"Straight angle" is not used in the UK, hence the possibility of misunderstanding. In this case, I don't think usage on either side of the pond is completely clear. In both cases further explanation of meaning is needed to avoid confusion between straight line and straight angle. I have no objection to "straight angle" being used in the article, since it is relatively easy for British readers to work out that it means an angle of 180 degrees. Dbfirs 20:16, 29 October 2009 (UTC)
However, "straight angle" is used in the US and elsewhere. These articles are not primarily for UK consumption and secondarily for others. The fact that it is not used in the UK should not affect its inclusion in an article. We need to avoid views which are UK-centric. Additionally, I would have to disagree with you that the the term "straight angle" is not completely clear on this side of the pond (a sweeping statement where you now speak for US schoolkids). From first hand knowledge, elementary schoolkids here find it no more difficult to understand "straight angle" than for them to understand "acute angle" or "obtuse angle". The definition is clear - an angle whose measure is 180°. The somewhat disparaging comment regarding US schools teaching confusing and self-contradictory math terms concerns me greatly. Confusing? Possibly to some, but not US schoolkids. Self-contradictory? It is definitely neither self-contradictory nor contradictory. However, the assertion that an angle of 180° is not an angle at all does appear to be self-contradictory. I do agree that British readers can easily understand the term "straight angle". JackOL31 (talk) 04:22, 2 November 2009 (UTC)
I thought that was what I'd said. I'm not really arguing with you on this. I was originally just making a comment about the original misunderstanding of the words, presumably by a US reader?. A straight line is not an angle until it is divided into two rays by a point. I agree that these articles are not primarily for UK consumption and secondarily for others, but nor are they the opposite. We should agree on the clearest usage for all of our readers. Dbfirs 08:38, 2 November 2009 (UTC)
I would say there was a deeper misunderstanding of the mathematics by the person placing the original post that was only exemplified by his misunderstanding of the term "straight angle". I say that with no disrespect. Regarding straight angles, I don't approach the concept of straight angles as you do. When I am thinking about a straight angle, I am picturing a terminal side of an angle moving counterclockwise and stopping where the protractor says 180°. I also picture a small semicircle centered over the vertex with a little arrowehead at the end of the arc where it meets the angle's terminal side. A line, per se, does not enter my mind. Regarding consumption, I believe my position has been to promote a more inclusive environment for these articles and not a one-sided viewpoint. I am going to assume you have reversed your position and you no longer consider the term to be self-contradictory. Perhaps your indictment of US schools has also been reversed? JackOL31 (talk) 02:10, 3 November 2009 (UTC)
I still claim that it is the line that is straight, not the angle, but yes, my original comment was just a rant based on the misunderstanding, and I agree that the term is not an unreasonable one (it is just not used in the UK, and appears contradictory to those whose first mental picture of an angle is acute), hence my assumption that it was the cause of the confusion, but you are probably right that the misunderstanding is deeper. I withdraw my comment ( - it was not intended as a serious criticism, and I did advise readers to ignore it when I originally made it!) Dbfirs 08:50, 3 November 2009 (UTC)
OK, we'll just leave it at that. As an aside, it is defined in the Cambridge Press site as, "An angle of 180°, i.e. a straight line. It is equal to two right angles." Can't say that I'm entirely pleased with the last sentence in the definition, but I'm going to stop tilting at windmills. JackOL31 (talk) 15:43, 3 November 2009 (UTC)
I'm not happy with the "straight line" bit either. The definition seems to have come from Syllabus of Plane Geometry, 1876, where the original says "When the arms of an angle are in the same straight line ...". Dbfirs 17:16, 23 November 2009 (UTC)

## Wording

I've tried to re-word the congruence section to make it both precise and easy to read. To achieve this, it seems better to reserve the word "congruent" for its principal sense (in two or more dimensions), and avoid its use for mere equality of length or turn. Other experienced editors seem to agree with this approach (see discussion above). Views (and further improvements in wording) are welcomed. Dbfirs 09:21, 13 February 2009 (UTC)

I'll admit that I have never heard that the word "congruent" has a principle sense of two or more dimensions, and that its use should be avoided for mere(??) equality in length or turn. On the contrary, Schaum's Outlines: Geometry, first published nearly half a century ago and still in publication (1998 4th Ed.) states, 3.3B - Two line segments having the same length are said to be congruent. In section 3.5D - Congruent angles are angles that have the same number of degrees. (I'm citing from the 3rd Ed.) Similar definitions are found in the Honors Geometry book for our school district. Could you cite the references where it is stated that the word "congruent" has a principle sense of 2 or more dimensions? JackOL31 (talk) 22:50, 7 December 2009 (UTC)

## Questions

Would someone please tell readers what program was used to draw the diagrams and write the equations, they are very well done.

## Just a thought

I think this article is very good. As a general reader i found it interesting and the supplementary images are fantastic. One thing that could be added is an overview of the history of the triangle i.e when did the triangle enter into a formal system of knowledge and why? How did ancient peoples percieve it's usefulness? Yakuzai 28 June 2005 22:02 (UTC)

## New formula for area of a triangle

I have added a new formula for the area of a triangle which I came up with when I was helping a student use the cosine rule to find an unknown angle for a triangle given its three sides and then proceed to find the area. The formula appears on another site but please feel free to verify it.

${\displaystyle {\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}$ - —Preceding unsigned comment added by 86.135.15.252 (talk)

## Use of the word "equal"

Whilst I understand the changes made by anon editor 72.178.193.150 to avoid the use of the simple word equal when referring to sides and angles, I think that the increase in precision is offset by a loss in clarity. Euclid used equal. British mathematicians use equal. Some American mathematicians use equal, but I don't know how many. What do others think? Can we reach some compromise that preserves clarity for the beginners who are most likely to need this article? Dbfirs 06:40, 13 July 2009 (UTC)

Could anon editor 72.178.193.150 please discuss this (and also learn correct prepositions)? Dbfirs 01:36, 30 July 2009 (UTC)
I've seen your reply here and other places regarding this topic and I thought I'd throw in my 2¢ on the subject. The meaning of the word congruent is dependent upon its usage (see www.mathopenref.com/congruent.html). Congruent line segments means having the same length. Congruent angles means having the same angular rotation. Congruent polygons means corresponding sides and angles have the same measure. The difference is that congruence is used when a numerical measure is not known or required. On the other hand, the measure of a line segment or the measure of an angle can be equal to some other line segment or angle since numerical values can be placed upon them. In a proof in the US, a lower case "m" would be placed in front of the angle symbol to denote measure, thus allowing the use of arithmetic operators (in other words, we can make equations). American geometry textbooks distinguish between "congruent" and "equal" and I see no loss in clarity. "Equal" is for numerical quantities which is why one doesn't say two triangles are equal, one says they are congruent. Having said all that, I wouldn't even begin to try to enforce that here. JackOL31 (talk) 23:09, 3 October 2009 (UTC)
Another example of "divided by a common language"? I'm glad Australians understand both usages. Dbfirs 23:26, 11 October 2009 (UTC)
I think mathopenref says it best: "Two objects are congruent if they have the same dimensions and shape. Very loosely, you can think of it as meaning 'equal', but it has a very precise meaning that you should understand completely, especially for complex shapes such as polygons."
Geometry textbooks introduce the concept of "congruence" because of the difference between the terms "equal" and "congruent". Otherwise there would be no need to introduce a new term - or for that matter, a new mathematical symbol! JackOL31 (talk) 01:51, 18 October 2009 (UTC)
I don't think "congruent" means "equal"! It has a much more precise meaning, as in our article on the subject. That's why I object to its use when just "equal" is intended (as for angles drawn different in dimension, but equal in rotation). Those who have studied Euclid are happy with this usage, though it seems that American school students are taught a different usage of the words. Dbfirs 09:36, 18 October 2009 (UTC)
I understand the different interpretations of congruence with respect to angles and I agree those different usages can lead to misunderstandings. Also, I understand your preference not to use "congruence" with respect to angles. Unfortunately, not using congruence when many expect to see congruence in and of itself leads to confusion. In addition, substituting "equal" for congruence when many again would expect to see "equal in measure" makes the matter even worse. While Euclid's use of "equal" for angles first presented 2300 years ago may make sense for some people, that usage for others in today's world is mathematically incorrect. Why is the nonusage of "congruence" and usage of "equal" with respect to angles preferred by the UK and others take precedence over the usages preferred by the US and others? I find it interesting that you acknowledge a different usage in the US (and elsewhere), but leave it at that. JackOL31 (talk) 21:37, 18 October 2009 (UTC)
From a UK perspective, I object to the use of congruent for two different representations of angles of the same measure when the representations are clearly not congruent. This is very confusing to students outside the USA where use of language is different. Our aim here is clarity for all our readers, hence the unambiguous "equal in measure" compromise. Dbfirs 19:53, 29 October 2009 (UTC)

From a US perspective, I object to the exclusion of terminology used in the US to maintain a UK-centic perspective. The following quoted words were used earlier on this discussion page: "On a fundamental level, we are not here at Wikipedia to decide what is true or not, but rather to report what others have said about things. "Verifiability, not truth" is the catchy phrase. If there are alternate definitions for fill in the blank in circulation, then those definitions should all be present on the fill in the blank page. If there are alternate definitions for fill in the blank, those definitions need to be present." You posted directly below those words without objection. Accordingly, I can cite numerous texts and math websites illustrating an alternate definition for congruent angles. To get the ball rolling, Schaum's Outlines: Geometry, 3rd Ed., copyright 2000 (originally copyrighted 1963) states, "Congruent angles are angles that have the same number of degrees. In other words, if m<A = m<B, then <A [congruent symbol] <B." Citing the mathopenref website: "Congruent Angles - Definition: Angles are congruent if they have the same angle measure in degrees." Changes will be necessary to this article and to the Congruence (geometry) article. Our aim here is clarity for all our readers, hence the inclusion of the other definition for "congruent angles" is necessary. For this page, I am not suggesting that "equal in measure" be replaced by "congruent angles", but rather words stating the alternate definition, the expectation by some to see the use of the term congruent angles, and an explanation regarding the possible confusion thus resulting in the "equal in measure" usage. The definitions need to be presented in a NPoV, matter-of-factly manner with no spin either way. JackOL31 (talk) 04:07, 2 November 2009 (UTC)

I'm told that the terminology is common in the USA only amongst "some educators". Nevertheless, I agree that we should aim for clarity for all of our readers, and obviously that includes the many USA students who have been taught this use of "congruent". Perhaps the main explanation should go at angle and at congruence (geometry), with just a brief note in the triangle article. Dbfirs 08:59, 2 November 2009 (UTC)
I cannot speak for all US educators, they may hold varying interpretations. What I do know is that all the elementary and high school texts I've seen (used in the US) agree with the definition stated by the Schaum's Outlines: Geometry. I agree that we should include the above definition for congruent angles. I also agree with your last statement and would add that the more appropriate place would seem to be Congruence_(geometry). The issue with the "Angle" page is that it precludes the meaning of angle as angle measure by only allowing angle figure. I would say one step at a time. First, Congruence_(geometry), then Triangle, and perhaps Angle. My next post will be to the Congruence_(geometry) talk page. JackOL31 (talk) 16:33, 3 November 2009 (UTC)
I've reconsidered my earlier thoughts and now believe it best not to make changes here. This is an article on Triangles, and a discussion of congruency would be out of place. Since one definition for "congruent angles" is "equal in measure" and that is the terminology used on this page, I believe that should be sufficient. JackOL31 (talk) 17:24, 7 November 2009 (UTC)

the measures of two angles of a triangle are given. 68* and 84*. whats the measure of the 3rd angle? —Preceding unsigned comment added by 71.184.158.30 (talk) 21:27, 22 February 2010 (UTC)

## Another error in a definition (isosceles triangle)

The definition of isosceles as applied to triangles (since at least the 1500s, and I think since Euclid, but someone who can read ancient Greek might check for me) is "having exactly two sides equal", not "at least two sides equal". I was surprised to see that both Wikipedia and Wiktionary had incorrect (by original definition) formal definitions of the word isosceles as applied to triangles. Is this an example of "divided by a common language", or just loose thinking by Eric W. Weisstein of Mathworld who seems to be the Authority on all things mathematical in the USA? (He is a much cleverer man than I, and I admire his collection of facts, but is he infallible, and is he the sole arbiter of the mathematical content Wikipedia? Perhaps he was influenced by categories of quadrilaterals where there are many subsets; whereas triangles are divided into three disjoint sets: scalene, isosceles OR equilateral.)
I intend to alter the Wikipedia article to include the formal Euclidean definition, but retain the modern (mis-used in my opinion) definition because some websites and texts use this. Which definition do American schools use? USA websites seem to give contradictory answers.
I can provide three quotes from early English Euclidean geometry to back up my claim. What does anyone else think? dbfirs 09:09, 17 January 2008 (UTC)

I've modified the usage notes, leaving the Eric W. Weisstein of Mathworld interpretation since some US text books seem to follow this, but I've put a link to Wiktionary where the citations of original usage can be seen. Is this OK? dbfirs 14:57, 17 January 2008 (UTC)
If an author decides to use a different definition than the one given by Euclid, it does not mean the new definition is wrong. The tendency in modern mathematics is to not exclude special cases from definitions, and in particular not if the theorems based on the more restricted definition typically equally apply to the special cases. If you Google the search term ["a triangle is called isosceles"], you will see that in present usage equilateral triangles are usually not excluded.  --Lambiam 17:06, 19 January 2008 (UTC)
Yes, I appreciate that many modern definitions follow Mathworld, though I can find other citations for the traditional division of triangles into three disjoint sets which I was taught (though that is nearly fifty years ago!). I am happy with your modification which preserves NPoV. Thanks. dbfirs 09:49, 25 January 2008 (UTC)
Euclid's wording for the equal sides of an isosceles triangle: δύο μόνας ("only two"). Here is Richard Fitzpatrick's rendering, from the Definitions in Elements:

20. And of the trilateral figures: an equilateral triangle is that having three equal sides, an isosceles (triangle) that having only two equal sides, and a scalene (triangle) that having three unequal sides.

So it is confirmed that Euclid partitions triangles three ways. If we take the meanings of equilateral and scalene (LSJ: "limping, halting, uneven") as given, it would be uncharacteristically inelegant for Euclid to pick out and name the conjunction of the remaining triangles and the equilaterals, with a single term isosceles. And he does not. If we were to include equilaterals as isosceles, to be similarly consistent we would have to include isosceles as scalenes. Ugly!
But we do live with such an ugliness for numbers, yes? Among real numbers, integers are rationals and algebraics, but they are not irrationals or transcendentals. That in itself is different from the Euclidean kind of classification; it is of a mixed kind, and it seems benign. But if we look at the article Quadratic irrational we see the kinds of imprecision that can arise if we are careless. A crucial clarification is missing from that article, though it is present at the linked [Mathworld article].
– Noetica♬♩Talk 11:02, 25 January 2008 (UTC)
Perhaps further elucidation is possible, such as pointing out certain consequences of the definition. But what is both crucial and unclear in our article? As to the MathWorld article, are the roots of x2 − 6x + 1 quadratic surds or not?  --Lambiam 17:31, 25 January 2008 (UTC)
From a standard formula, the equation x2 − 6x + 1 = 0 has these roots:
${\displaystyle {6\pm {\sqrt {32}} \over 2}}$
Reduce these to:
${\displaystyle {3\pm {\sqrt {8}} \over 1}}$
(The reduction is obviously necessary for the analysis below, but this information is not provided even at Mathworld.)
8 is not a square-free integer, since its prime decomposition contains repeated factors. Therefore, from the (questionable!) definition in the Mathworld article, these roots would not be quadratic surds.
However, the Mathworld definition appears to be not only questionable but wrong (because it is fatally incomplete). In fact the roots are quadratic surds. From this source, the correct formulation (I hope!) goes like this:
A number is a quadratic surd if and only if it can be expressed in this way:
${\displaystyle {P\pm \ R*{\sqrt {D}} \over Q}}$
where P, Q, R, and D are integers [Q > 0, R > 0, I would add], D > 0 and not divisible by a square [other than 1, I would add], and Q is a divisor of
${\displaystyle {P^{2}-R^{2}*D}}$
In our example, we reduce further:
${\displaystyle {3\pm \ 2*{\sqrt {2}} \over 1}}$
And indeed, the roots are quadratic surds. That's what I would have thought all along, before Mathworld and the WP article led me astray! At the very least, Mathworld ought to include R in the definition, and it ought to give those restrictions on the integers Q and R that I supply.
I will not attempt to fix Quadratic irrational, which has a few really serious errors, since it is not my area.
Lambiam, thank you for assisting in my education, belated though that may be. I'll have you know I got up at 4:00 am to attend to this, since I was somnolently pretty sure that not all was kosher. Have I got things right now? I suspect that, if I have things wrong (as I still fear), then one or more of the sources outside of WP that I have looked at also have it wrong.
I do intend to copyedit the present article, though, when I have time. It isn't bad! But it's fundamental, and should therefore be kept polished. I think that more about the non-Euclidean sense of isosceles should be shifted to the note. [Done.]
– Noetica♬♩Talk 10:09, 26 January 2008 (UTC)[Substantially amended contribution, yet again– Noetica♬♩Talk 03:40, 27 January 2008 (UTC).]
Rather than looking to Euclid's definition of isosceles triangles to determine whether an equilateral triangle is isosceles or not, you can look to see whether equilateral triangles exhibit the same properties as isosceles triangles. If you look at the general isosceles diagram above, you will see that isosceles triangles have the properties of two equal sides and two equal base angles. Given an isosceles triangle, the Triangle Inequality Theorem states that the third side must have a length greater than zero but less than twice the length of an isosceles side. So, starting with the isosceles diagram above, one can change the third side to a length equal to that of the isosceles sides. Since we are starting with the isosceles diagram and changing the length to a length greater than zero but less than twice the length of the isosceles side, nothing has changed with respect to isosceles. What has been shown is that an equilateral triangle has the same properties as an isosceles triangle and therefore is isosceles. An analogous procedure can be performed by noting that the angle joining the equal sides must be greater than 0 degrees but less than 180 degrees. Again, starting with the isosceles diagram above and setting the angle joining the isosceles sides to 60 degrees, an equilateral triangle results and the isosceles properties again have not changed. Lastly, if we were to define acute, right and obtuse isosceles triangles, then an acute triangle having a 60 degree angle would not be allowed to be a member of the acute isosceles triangle set. I would say this is a mathematical contradiction and despite the meaning or intended meaning of the word "isosceles", an equilateral triangle still ends up as isosceles.  JackOL31 (talk) 01:13, 6 May 2009 (UTC)
You can't prove a definition. An equilateral triangle can only be isosceles if you define isosceles inclusively. This is a valid option, but I was taught to divide triangles into three disjoint sets: equilateral, isosceles and scalene. If the terms are inclusive, then it makes the word "scalene" pointless because all triangles would be scalene. Dbfirs 07:11, 30 June 2009 (UTC)

I think that Euclid was trying to classify triangles based on the length of the THREE sides, not just two. Scalene triangles have the length of all sides different, isosceles only two and equilateral all three. The point is that all sides are considered in this classification.

Today the classification is done in terms of the number of sides of the same length, because in practice (and that means, in terms of writing theorems for the theory), the type of theorems that are proved for triangles with two sides of the same length do not depend on the length of the third side, so it is a moot point if the theorem that is proved for a triangle that has two sides of the same length is equilateral or isosceles in Euclid's definition. Today it is more convenient to call all these triangles isosceles, that is, to include the equilateral triangle as a particular case of an isosceles triangle, since the theorem that is proved for them is true for an equilateral triangle. —Preceding unsigned comment added by 72.178.193.150 (talk) 03:33, 13 July 2009 (UTC)

Nothing moot about Euclid! He was clear in his definitions. I agree that, in the USA, the definition of the word isosceles seems to have changed (is this uniform throughout North America?), but elsewhere Euclid's definitions have remained in common use for 2000 years, and are taught in British schools. Dbfirs 06:26, 13 July 2009 (UTC)
What I said was that a theorem that applies for triangles with two sides of the same length applies to a triangle with three sides of the same length. It is not relevant the knowledge about the third side. Today is common to call those triangles isosceles because it is irrelevant if the third side is different or not in length, and certainly all isosceles triangles in the Euclid's sense are isosceles in this sense. The point is that Euclid was thinking of THREE sides when classifying triangles, and today we do not see the need to do that, so we call triangles with two sides of the same length isosceles. I am not quite sure why you keep arguing about this. Take a look at the proof of Theorem 4 in Book I of the Elements, and try to find the error. Today we recognize that that theorem is actually a postulate, not a theorem. That is certainly a very important point, much more than this minor point. Euclid was not wrong in this point, nor we are wrong. Today it is more convenient to call isosceles triangles that have at least two sides of the same length, than to do otherwise, since the theory is much cleaner this way. This is not a regional convention, but a matter of convenience. —Preceding unsigned comment added by 72.178.193.150 (talk) 01:19, 14 July 2009 (UTC)
Yes, I agree that there is some logic in using an inclusive definition so that proofs automatically carry forward to equilateral triangles, but there is also a clean logic in dividing triangles into three disjoint sets (following Euclid), and I believe that this is more common outside the USA. Both conventions are valid. Both conventions are used. They are just alternative definitions. Dbfirs 08:10, 14 July 2009 (UTC)
First, the statements, "...I agree that there is some logic in using an inclusive definition so that proofs automatically carry forward to equilateral triangles..." and "...there is also a clean logic in dividing triangles into three disjoint sets..." are contradictory. Either equilateral triangles are a subset of isosceles triangles or equilateral triangles are a disjoint set from isosceles triangles. You cannot have it both ways. Secondly, we must obey the rules set forth by Algebra of Sets (Set Theory). What is common or uncommon is not at issue here. What Euclid states is not at issue here. One does not have a free hand with mathematical definitions. If a definition states that certain triangles are disjoint from others, then they must be mathematically disjoint. One does not have the option to say triangles are disjoint from others by definition! The definition fits the mathematics. Now, isosceles triangles have two equal sides (let's say of length s) joined at a vertex and whose third side varies in length from greater than 0 but less than 2s (as dictated by the Triangle Inequality Theorem). I maintain that equilateral triangles share the same isosceles properties as nonequilateral isosceles triangles. To be disjoint, one must show an isosceles property lost by having a base length of 1s (it can have additional properties, it just cannot have fewer properties). Maybe it's no longer a triangle? Maybe it no longer has base angles equal? Maybe the sides opposite those base angles are no longer equal? Can one show that an isosceles triangle with a base of 1s has any fewer isosceles properties than those with a base of .5s, .999s, 1.001s or 1.314159s? I think not since nothing magically happens somewhere between .999s and 1.001s with respect to isosceles. One cannot make the claim that it is disjoint simply because "1s" is disjoint from "different from 1s". One could make the same claim for any valid base length. Recall that for all isosceles triangles, two equal sides (and consequently equal base angles) are a given. Only the length of the third side is variable. Since you are claiming that equilateral triangles are disjoint from isosceles triangles, it is incumbent upon you to show which isosceles property is lost when the third side has a length of 1s (equal to the isosceles sides).  JackOL31 (talk) 22:51, 26 July 2009 (UTC)
Please check out the following UK websites: http://mathsblog.co.uk/wp-content/uploads/2008/04/5501-05.pdf and problem 3a located at the bottom of this webpage http://openlearn.open.ac.uk/mod/resource/view.php?id=328482  JackOL31 (talk) 23:23, 26 July 2009 (UTC)
Of course my alternative statements are contradictory because they comment on two different alternative definitions of the word isosceles. One definition has been in use for 2000 years, the other is the invention of modern mathematicians, especially in the USA (and, I agree, also on some UK websites), influenced by the inclusive definitions for quadrilaterals. Both definitions are valid, provided that they are consitently used, but the webpage you quote [2] mixes the definitions because it uses my (three disjoint sets) definition at the end of example 7. I don't think we will ever agree on which definition is "best", so perhaps we can just agree that there are different usages, and some confusion by those who don't think clearly about their definitions. Dbfirs 08:32, 27 July 2009 (UTC)

Whew, many things to address here. First, you stated, "...An equilateral triangle can only be isosceles if you define isosceles inclusively..." With all due respect, this is not mathematically correct. One merely defines sets of objects, whether those sets are a subset or disjoint from another set is determined by its mathematical properties, not by definition. Scalene and isosceles were defined as subsets, they are disjoint because the members of each subset have unique properties beyond the triangular properties.

Secondly, you indicated, "...If the terms are inclusive, then it makes the word "scalene" pointless because all triangles would be scalene." Mathematically, that is definitively false. The set of all triangles having 0 equal sides does not include the set of triangles having 2 equal sides. However, the set of triangles having 2 equal sides (the third side being equal or unequal) does intersect the set of triangles having 2 and 3 equal sides.

Thirdly, you stated, "...I agree that, in the USA, the definition of the word isosceles seems to have changed (is this uniform throughout North America?), but elsewhere Euclid's definitions have remained in common use for 2000 years, and are taught in British schools." This appears to be conjecture and misinformation. A definition contradicting Euclid's definition has been used throughout the world for hundreds of years. In addition from my research of UK websites, it appears that UK primary schools teach that isosceles triangles "do not include equilateral", "include equilateral" and "no specific reference either way". Note: the "common use for 200 years " topic is addressed later in this discussion.

Regarding the topic of equilateral triangles as a subset of isosceles triangles or disjoint from isosceles triangles, you conveyed the proposition, "...Both conventions are valid...". This is mathematically incorrect. If one has a proper subset of a parent set, the subset cannot be disjoint from the parent set, by definition of a proper subset. You are not allowed to violate the rules of Algebra of Sets. The set of triangles having only 2 equal sides is disjoint from the set of triangles having 3 equal sides (more clearly, 2 and 3 equal sides), but they are both subsets of the set of triangles having two equal sides (no claim regarding the third side). In a later post, you claimed that, "...Both definitions are valid, provided that they are consitently used...". Again, this is a violation of the Algebra of Sets since the same members are involved and the set of equilateral triangles is first a member of, and then disjoint from, the set of isosceles triangles.

Your reply that, "Of course my alternative statements are contradictory because they comment on two different alternative definitions of the word isosceles...". I believe you have misread my statement. I was noting the fact that your claim for both contradictory sides as valid does not have merit, mathematically speaking. You are trying to have it both ways. [Please note the earlier statements regarding Algebra of Sets]

You mentioned, "...One definition has been in use for 2000 years, the other is the invention of modern mathematicians, especially in the USA (and, I agree, also on some UK websites), influenced by the inclusive definitions for quadrilaterals." Much of this is simply conjecture on your part with a hint of both personal and mathematical bias. The notion that mathematical rules are not consistent and differ for triangles or quadrilaterals, well, speaks for itself. As noted earlier, I will address the, "...in use for 2000 years", statement later.

Also, you indicated that the example just prior to Example 8 in the OpenLearn link was a "mixing of definitions". This appears to be a grammatical misinterpretation on your part of the sentences, "In an isosceles triangle, two sides are of equal length and the angles opposite those sides are equal. Therefore, (base angle) alpha = (base angle) beta in the triangle below." The above actually conveys the meaning that an isosceles triangle has two equal sides and makes no claim to what the third side can be or cannot be (and in the above case, no claims for the third angle, either). The above is NOT Euclid's definition of isosceles triangles, although it is often mistaken for it. Since the author's statements do not place restrictions on the third side (or angle), the problems given are consistent with the definition offered. The author's subsequent problems were simply pointing out two of the more interesting isosceles triangles: equilateral and right isosceles.

It appears that much of your claim to wide and historical usage of Euclid's definition is based on the false premise that the definition you have read was Euclid's definition of isosceles triangles. If a definition does not explicitly mention the concept of "only" or "exactly" two equal sides, then it is not Euclid's definition. It is a simple matter of grammar, and there is a tremendous difference between "Isosceles triangles have only two equal sides" and "Isosceles triangles have two equal sides". You may also see it written as, "An isosceles triangle has two of its sides equal." I will now cite some examples of isosceles definitions contradicting Euclid's definition from my collection of 19th and 20th Century math textbooks:

Mathematics, Compiled from the Best Authors, and intended to be the TextBook of the Course of Private Lectures on these Sciences in the University of Cambridge, Second Ed., Samuel Webber, President of the University at Cambridge. Printed at the University Press, 1808

25. An isosceles triangle is that, which has two equal sides.

The Normal Geometry: Embracing a Brief Treatise on Mensuration and Trigonometry, Edward Brooks, Christopher Sower Company, 1865 (Recopyrighted 1884)

22. An ISOSCELES TRIANGLE is one which has two of its sides equal.

New Plane and Solid Geometry (Revised Edition), G. A. Wentworth, 1888, Ginn & Company Publishers, 1893

129. A triangle is called, with reference to its sides, a scalene triangle ...; an isosceles triangle, when two of its sides are equal; an equilateral triangle when its three sides are equal.

Plane Geometry and Supplements, Walter W. Hart, Veryl Schult, Henry Swain, D.C. Heath and Company, 1959

Triangles--Congruence

59. (b) An isosceles triangle is a triangle having two equal sides.

Theorems Based on Parallels--Isosceles and Equilateral Triangles

116. If two angles of a triangle are equal, the sides opposite them are equal and the triangle is isosceles.

It is extremely important to note that equilateral triangles are not excluded from any of the above isosceles definitions, unlike the main definition currently offered on Wikipedia.

Lastly, I would like cite online definitions which I would consider to be extremely authoritative references on the matter. Note: I have replaced the periods in the url with underscores to prevent the website from being hyperlinked.

1) From the Oxford Press Concise OED (in association with Oxford University):

adjective (of a triangle) having two sides of equal length.

2) From a site sponsored by the Cambridge University Press (in association with Cambridge University):

thesaurus_maths_org/mmkb/entry.html?action=entryByConcept&id=73&langcode=en

A triangle which has two equal sides. The third side is called the base.

It is striking how the definitions given by Oxford Press and Cambridge University Press are markedly different than yours posted on Wiki. You can not help but notice the extreme care used by them to avoid giving the impression that isosceles triangles have "exactly" two equal sides. One has to ask why they didn't include the word "exactly" in their definitions, as simple as that would have been. Also, they stated that the third side is called a base, not that the third side is unequal from the isosceles sides. It is also worth noting that the Cambridge University Press link allows you to create various isosceles triangles without any disclaimer when an equilateral triangle is formed.

The definitions stated by both Presses are correct, definitions based on the complete properties of isosceles triangles. They are not based on counting the number of equal sides since 2 equal sides subsumes 3 equal sides. JackOL31 (talk) 23:45, 1 August 2009 (UTC)

I agree that the definitions stated by both presses are correct, but I maintain that they are based on counting sides. They carefully avoid the controversy, as do the other publications that you quote, by specifying neither "at least" nor "exactly". I totally reject your claim of some kind of inevitable mathematical "logic" argument here. Mathematicians are allowed to change definitions if they wish, but there is no inevitability about doing so, just agreement (or the lack of it). You might be interested in the following:
• c 300 BCE Euclid Elements, Book I, Definition 20, English translation,
Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
• 1551 Recorde, Pathways to Knowledge,
There is also an other distinction of the names of triangles, according to their sides, whiche other be all equal..other els two sydes bee equall and the thyrd vnequall, which the Greekes call Isosceles ...
• 1570 Billingsley, Euclid Book 1, Definition xxv,
Isosceles is a triangle, which hath onely two sides equall.
• 1571 Digges, Pantometria I,
Isoscheles is such a Triangle as hath onely two sides like, the thirde being vnequall, and that is the Base.

Definitions were clear in the past, before modern "Algebra of Sets" mathematicians started tinkering! Dbfirs 10:38, 2 August 2009 (UTC)

It is very difficult to discuss a topic when you do not respond to many of my discussion points. I have also noticed that I am using mathematics extensively and you are using very little mathematics or none at all. So, I will have to assume once more that you now agree to the validity of my other arguments except for the ones mentioned in your last post. I will then limit my responses to those points you brought up in your last post.
Your statement, "...but I maintain they are based on counting sides...". That is conjecture on your part. There is no supporting evidence on either site that they are limiting isosceles to two equal sides. In fact, the mathsorg site explicitly allows it. If the definitions were to explicitly limit isosceles to 2 equal sides, then they could have been clearly stated with the wording "having exactly two sides of equal length" and "A triangle which has exactly two equal sides". The grammar of the definitions clearly does not exclude equilateral triangles from any of those I listed previously, either from the books or the websites. This is an observable fact.
Your claim, "They carefully avoid the controversy, as do the other publications that you quote, by specifying neither 'at least' nor 'exactly'" is conjecture on your part. You cannot speak for those authors I cited, other publications or the authors of those websites. You are making unsubstantiated claims. You are essentially offering as a supporting argument, "They said this, but I say they meant this other thing". I maintain we must follow the actual wording and not wording that is not actually present. Also, I am not aware of any controversy that existed in 1808?
You state, "I totally reject your claim of some kind of inevitable mathematical "logic" argument here." I actually do not know how to respond to this statement. The rejection of mathematical "logic" arguments has no place in this discussion and is indicative of a position that cannot be mathematically nor logically supported.
You wrote, "Mathematicians are allowed to change definitions if they wish,...". Not true, they must support any change with mathematical arguments and not have it rejected, i.e. shown to be mathematically false. Definitions are changed when the previous definition proves to be mathematically lacking or incorrect. They are not changed on a whim as your statement seems to indicate.
I am aware of Euclid's Elements and the time it was written. One cannot confuse being around for a long time with being widely accepted for a long time. The notion that the sun and the planets orbit the earth has been around for a long time. I agree that those publications either cite Euclid's work specifically or repeat his work. However, his definition was rejected at some point in time and a new "corrected" definition arose. I cannot say when, I am not a math historian and I am assuming that you are not either. What I can say is, that at least as early as 1808 (and probably earlier) the word and thus the concept of "only" or "exactly" was dropped from the definition. This is not to say you can't still find Euclid's definitions in some book or on some website, but by checking numerous books and websites the greater proponderance of acceptance points to equilateral triangles being accepted as a subset of isosceles.
You cite the following definitions which I will excerpt in part, "...has two of its sides alone equal...", "...els two sydes bee equall and the thyrd vnequall...", "...hath onely two sides equall..." and "...hath onely two sides like, the thirde being vnequall...". This bolsters my point that for equilaterals to be excluded from isosceles, it is necessary to explicitly exclude it from the set. The omission of this explicit exclusion in subsequent definitions cannot be construed as the same meaning. You cannot logically or grammatically say that using the word "only" or not using the word "only" conveys the same meaning, except when the word is used superfluously.
You mentioned, "...before modern "Algebra of Sets" mathematicians started tinkering!" Algebra of Sets has its basis in Set Theory and here is what Wiki say regarding Set Theory, "Set theory, ..., is the most common foundational system for mathematics. Beyond its use as a foundational system, set theory is a branch of mathematics in its own right, ...". On a wiki math web page, or anywhere else for that matter, you cannot dismiss a branch of mathematics simply because it does not support your position. Any arguments based on that premise must be discarded.
There is really only one property of isosceles triangles, a triangle having two equal sides (and it can be shown that the angles opposite those sides are equal). When you generate all the possible triangles formed by starting with two equal sides, if two of those triangles formed share isosceles properties, then they all do since they all came from the same two equal sides. (Imagine bending a drinking straw in half and standing it upright on a table, then slowly spread apart the ends until the straw is lying horizontal on the table.) The only constants are the fact that the upright sides are equal and the angles opposite those sides are equal. Everything else variable. Definitions define the constant properties, not the variable ones. The variable properties are simply allowed to vary, in this case the length of the base and the measure of the angles. You can only geometrically state it is not isosceles if one of those constant properties is not maintained. I have now shown that they are maintained using geometry and not Algebra of Sets or Set Theory (which are still valid to use). The definition you support has been mathematically refuted.
I would appreciate any mathematical discussion prior to my revising the parenthetical following the isosceles definition. My intent is to state that the definition includes equilateral triangles and reference Euclid's definition of using "only" as still in use but no longer widely accepted. JackOL31 (talk) 14:53, 2 August 2009 (UTC)
The definition that I support has been mathematically refuted as a definition only relatively recently, and only by some mathematicians. I agree that the properties of isosceles triangles logically carry over to equilateral triangles, but in the world outside mathematics, two does not usually mean at least two, and the division of triangles into three disjoint sets (saclene, isosceles and equilateral) has two thousand years of usage, despite what some mathematicians might wish. Dbfirs 17:14, 2 August 2009 (UTC)
... (later) ... Just to clarify: I do accept the validity of your argument from an Algebra of Sets viewpoint, and I apologise if I seem to be ignoring your argument. Those of us who were taught (and have taught) only Euclidean geometry from a traditionalist viewpoint still use Euclid's definition "δύο μόνας" ("only two") sides equal. I think our problem is one of semantics. Could I ask whether, in set theory, the set of polygons having two diagonals is a subset of the set of polygons having five diagonals? Dbfirs 20:39, 2 August 2009 (UTC)

I guess I have a different take on the mathematician statement. It's one thing to use the Pythagorean Thm, it's another to prove it. Does the mathematician have the tools to make the call? Regarding "two" and "at least two", you're correct is saying that it is different in mathematics than in general conversation. The main properties (not definition) of a parallelogram are opposite sides equal and opposite angles equal. But it doesn't mean "only" opposite sides equal and "only" opposite angles equal since a square is a parallelogram. A square has opposite sides and angles equal (necessary for the set parallelogram) AND 4 sides and 4 angles equal (necessary for the subset square). Two pair of equal sides subsumes four equal sides. Always bear in mind that a definition is the absolute minimum properties that allows one to say, "Hey, I'm a member!" Never read more into it than what is said, never say more than what is needed. Regarding your semantics statement, we were talking about different sets and I kept meaning to clarify that. If you'll bear with me, let's say the set I = set of all triangles having two equal sides (if you see two equal sides, throw it in the pot). Let's say the set E = set of all triangles having three equal sides (really having two equal sides AND three equal sides). Let's say the complement of a set are all the members outside the set. Then what you have been calling the set of isosceles is I \ E, or the intersection of the entire set I with the complement of E. If you recall the Venn Diagram of a smaller circle contained completely inside the larger circle, then draw horizontal lines throughout the entire circle and vertical lines outside of the smaller circle. The result is a "doughnut", or the set of isosceles triangles with the equilaterals subtracted out. This is actually a relative complement, a complement of E but going no further than the set I. Whatever the names you call them, the important mathematical concept is that the set I \ E and the set E are part of the larger set I. So yes, what you call isosceles is disjoint from equilateral, but they are both subsets of uber-isosceles. (If we call I \ E isosceles, then we need to call I something such as uber-isosceles.) However, I would call the set I isosceles, and the set I \ E has no name, it's just the set of isosceles with the set of equilaterals subracted out. Maybe we want to call the set plain-isosceles. This is similar to subtracting out the union of rhombuses and rectangles (which includes the set of squares) from parallelograms. All that is left are the plain parallelograms, but they have no particular name. Regarding subsets, short answer is no since quadrilaterals (set of polygons having 2 diagonals) are disjoint from decagons (set of polygons having 5 diagonals). I do realize how difficult this is for you. Would you be open to working with me on new verbage for the definition? I must warn you, I have one more bomb to drop. Although I believe you will agree with me. JackOL31 (talk) 02:05, 3 August 2009 (UTC)

Just to clarify - difficult to accept a new definition, different from what you've held for a long time. JackOL31 (talk) 02:24, 3 August 2009 (UTC)
I loved the concept of uber-isosceles, and yes, of course I agree with your subset analysis, except that I expected you to say that the set of polygons having five diagonals (pentagons have exactly five, decagons have 45) is a subset of the set of polygons having two diagonals, because the pentagons (and all other polygons) satisfy the minimum property of having two diagonals.
I will be happy to agree on an adjustment to the wording in the triangle article, provided that you will allow mention of two thousand years of Euclid's usage (I can provide further cites), and provided that you are not a sockpuppet of anon editor 72.178.193.150. Dbfirs 09:54, 3 August 2009 (UTC)
First, please realize that uber-isosceles is not consistent with its closest cousin, the parallelogram. There is no uber-parallelogram. Also, I cannot say that an equilateral triangle is any less isosceles than any other of its brethren. That would be denying what I just proved geometrically. Equilateral triangles are first and foremost isosceles (having 2 equal sides) and then they are a subset of isosceles (having 3 equal sides). We must not forget what the geometry showed us, the only defining criteria is having 2 equal sides. The base can be any valid length and plays no part in determining an isosceles triangle. (1) Regarding the rewording, the main effort is to get it in line with the definitions given by the Presses (Oxford and Cambridge University Presses). The definition already given is actually acceptable, just the parenthetical needs to change to clarify that it includes equilateral triangles since they carry the same properties. (2) Another change is to state Euclid's definition and use throughout the ages but that is proven to have mathematical contradictions. As an aside - your proposal is fine with me but I think you tilting at windmills. No one is challenging its use throughout the ages, just that it has been shown to have mathematical contradictions. (3) There is a current trend to say that the definition of an isosceles triangle is a triangle having at least 2 equal sides. This is where I strongly disagree what is displayed currently and what Anon also indicated. That is NOT a valid definition. Rapid fire reasons: (a) one does not mention properties particular to a subset in the definiton of a parent set - only what it has, not what it could have, e.g. an isosceles triangle COULD HAVE 3 equal sides, well a parpallelogram COULD HAVE 4 equal sides but it doesn't belong in its definition (b) definition is backwards - first you state the overarching property, then you define subsets within the set, you don't say the set is a compilation of 2 subsets, Isosceles is not defined as nonequilateral isosceles triangles OR equilateral isosceles triangles anymore than triangles are defined as scalene triangles or isosceles triangles - or even uglier, at least scalene! (c) Since you are overriding the subsuming, the definiton becomes an isosceles triangle has only two equal sides or an isosceles triangle has three equal sides, the second clause is unconditionally false, it is only conditionally true ("could have") (d) if we can't use the base to exclude it from isosceles, we can't use it to include it either. Bottom line: any definition which explicitly mentions the third side of an isosceles triangle is wrong (it plays no part). I know if we don't say something, someone will want to replace the Presses definition with the uber-awful definition. JackOL31 (talk) 00:30, 4 August 2009 (UTC)
But you identified the set of polygons having two diagonals with the set of quadrilaterals. Surely, by your inclusive mathematical standards, the latter is a proper subset of the former? Dbfirs 08:51, 4 August 2009 (UTC)

Actually, I never said anything about a proper subset. In fact, I said quite the opposite. The set T is contained in set Q. The set Q is contained in set T. Then, set T = set Q. Set T is not a proper subset of Q and set Q is not a proper subset of T. A proper subset is when a set's members are entirely contained in another set but is not equal to the other set. We shouldn't take up space with this discussion. I don't think it's prudent to continually go over this, the geometric fact has already been proven. It's time to make the necessary updates, correct? JackOL31 (talk) 21:48, 4 August 2009 (UTC)

Are you afraid to discuss the consequences of inclusive definitions? Dbfirs 22:38, 4 August 2009 (UTC)
The situation is that you always have a question even if it is not well founded, mathematically. I've given approximately 23 arguments, all refuting statements you have made. There is no downside to including equilateral triangles in isosceles triangles. It fills in the holes that Euclid left, and we have already gone over all of this before. You are not bringing up any new issues and I don't chew my cabbage twice. www_mathpages_com/HOME/kmath392_htm  JackOL31 (talk) 01:38, 5 August 2009 (UTC)
It seems to me that you are not prepared to discuss the logical consequences of your mathematical position. The whole question is one of the meaning of your definition and the logical consequences of your interpretation. Might there be a contradiction somewhere? [3] Dbfirs 06:57, 5 August 2009 (UTC)
This, of course has nothing to do with the topic at hand. The consistency of arithmetic or not does not invalidate Algebra of Sets, if that is your point. Instead, I again present to you where does equiangular triangles go? Consider dividing isosceles into three disjoint sets: acute, right and obtuse. As you're dividing up all the isosceles triangles by changing the vertex from greater than 0 degrees to less than 90 degrees, a vertex of 90 degrees, and a vertex greater than 90 degrees but less than 180 degrees, you notice a problem. The arbitrary exclusion definition means that you will need to pull out equiangular triangles from the acute isosceles triangles and form another group within the set of isosceles. But wait, now you must once again pull the set out of the set of isosceles and place it at the same level as its grandparent set. Other than the clear mathematical fallacy of a set hopping up 2 generations, there is still have no clear mathematical explanation of its crime necessitating the removal from the set of acute isosceles triangles in the first place. One needs to look no further than the exclusive interpretation of isosceles for contradictions. No such contradictions from the correctly interpreted definition of isosceles triangles. I have refuted your claims 24 times, please do not presuppose that I am not prepared. JackOL31 (talk) 05:05, 6 August 2009 (UTC)
But I was trying to show a possible contadiction (logical and semantic rather than purely mathematical) in the inclusive definition. Please remember that Wikipedia is a general encyclopaedia, read by people of all disciplines. Dbfirs 09:26, 6 August 2009 (UTC)
In support of the (historically correct) definition that an isosceles triangle is on with exactly two sides of equal length, I point out the following:
Theorem. An isosceles triangle has only one line of symmetry.
This theorem clearly becomes false if one takes as definition that an isosceles triangle has at least two sides of equal length (for an equilateral triangle has three lines of symmetry). Symmetry is an important concept in geometry, since the number of symmetries of a figure form a group. In the case of an iscoceles triangle the group of symmetries is a cyclic group of order 2, generated by reflection about the line of symmetry; on the other hand, in the case of an equilateral triangle, the group of symmetries is isomorphic to the symmetric group of degree three (containing six elements - the three reflections in the three lines of symmetry, the two nontrivial rotations through 120 degrees, and the identity element). IMHO, it is better to stick with the historically correct definition of isosceles and point out that there is a West Pondelian trend (even if misguided, and largely due to educators, not mathematicians, who all too frequently disregard or are ignorant of tradition) to supplant it with the one that includes equilateral. -- Chuck (talk) 17:36, 6 August 2009 (UTC)
There is no logical contradiction, the drinking straw proved that. In fact, the straw proved that leaving equilateral triangles out IS the contradiction, geometrically speaking. Not being consistent with quadrilaterals is also a contradiction. Again, there is no semantic contradiction. The property of isosceles triangles is having 2 equal sides (Presses) with the angles opposite those sides equal. However, any definition that does not include all triangles meeting those properties is simply describing a subset. I understand Euclid's isosceles triangle definition, but since we can show that equilateral triangles have all the properties of the other isosceles triangles, Euclid's definition is lacking. Besides, the equiangular argument is irreconcilable. It's tantamount to saying, for example, that scalene triangles where the Pythagorean Theorem can be applied is no longer scalene. That would be a tough sell in and of itself, however kicking it up one more level and saying it's not a triangle is beyond reason. The mathematical reality is that equiangular is a subset of acute isosceles and acute isosceles is a subset of isosceles.
Regarding the general encyclopedia comment, I think consistency with the straightforward Oxford and Cambridge University Press definition of 6 to 7 words is reasonable. Now I need to get back to the general Wiki on triangles to learn more about 2-simplex (Polytopes), Menelaus' theorem, orthocenters, incircles & excircles, symmedians and Feuerbach points!

Regarding Chuck's comment, I did not ever read in geometry that symmetry overrides all other properties. I guess I'd be more inclined to put stock into what you state if those comments were also on the quadrilateral page. You'd have to convince many that a square is not a rhombus, not a rectangle and not a parallelogram. Extremely difficult since it can be shown that a rhombus has all the same properties as a parallelogram plus more, likewise for a rectangle and a square has all the properties of a parallelogram, a rhombus, a rectangle and more. This flies directly in the face of Euclid's historically correct(???) statement: "Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled;..." By focusing strictly on number of eqaul sides, Euclid missed the fact that certain shapes subsume other shapes. JackOL31 (talk) 04:07, 7 August 2009 (UTC)

Rather to my surprise, the American website www.math.com has this definition: "An isosceles triangle has just two equal sides ..." Dbfirs 08:19, 16 August 2009 (UTC)
Don't be surpised, there are others. Sites I've found to be very good are www.mathopenref.com, thesaurus.maths.org, and mathworld.wolfram.com. Most 5th Grade curriculums have the requirement that the student must know, "All equilateral triangle are isosceles, but not all isosceles triangles are equilateral." JackOL31 (talk) 21:28, 16 August 2009 (UTC)
Sometime in the future the Isosceles Triangle section shbe updated since there are 3 definitions in use. Definitions: 1) only two equal sides (Equilateral excluded), 2) two equal sides (Equilateral included), 3) "at least" - poorly worded rephasing of the 2nd definition. In Geometry & Trigonometry, The Colliery Engineer Company, copyright 1893, 1894, 1895, 1897 and 1898; p8, 40. An isosceles triangle, Fig. 27, is one having two of its sides equal. Followed by, 41. When the three sides are equal, as in fig. 28, it is called an equilateral triangle. An equilateral triangle is also isosceles.
Just wanted to share an early example I found today explicitly indicating the relationship. I think we're both in agreement that 3 definitions are in use. Would you like to work something up for an update? JackOL31 (talk) 01:05, 9 November 2009 (UTC)
Yes, I would even say four definitions, adding 4) two equal sides (Equilateral excluded), but I don't think it would help clarify the article to include all four. Perhaps an explanation from the historical viewpoint would be better. The whole argument is over whether triangles ar classified according to their number of equal sides, or according to (inclusive) properties. Dbfirs 09:17, 9 November 2009 (UTC)
I'm only seeing three, perhaps I'm miscommunicating? 1) Euclid's isosceles - no equilateral, 2) Having two sides, "not only" - equilateral included, 3) Using "at least" - same definition intent as previously stated, but poorly worded. I believe that's it unless I've missed something. If I have, please explain further about the fourth case. Regarding classifications, I wouldn't look at it that way. I see it more from a Set operations/Venn Diagram perspective. Does the set of triangles having 2 equal sides (in measure) need to exclude the set of triangles having 2 equal sides AND 3 equal sides? Do we exclude triangles because an adjacent side to the sides of interest or an adjacent angle to the angles of interest are also equal? To illustrate the point further, do we exclude two intersecting perpendicular lines from the set of vertical angles because only the vertical angles are equal and not the angles adjacent to them (i.e. vertical angles equal AND adjacent angles equal)? Note: AND is upper case to denote logical operation.  JackOL31 (talk) 01:38, 13 November 2009 (UTC)
The extra one comes from splitting 2 into 2a) two equal sides (Equilateral included) (your interpretation), and 2b) two equal sides (Equilateral excluded) (my interpretation), but we've been over all this before and we will just have to agree to differ. You use the words as they are (correctly) used for quadrilaterals to produce inclusive definitions; I use the words as Euclid used them to divide triangles into three disjoint sets: Equilateral, Isosceles and Scalene with no overlap. I believe that my usage is the normal one in the UK; you believe that your usage is the only possible one from an algebra of sets viewpoint (and I do see your point of view - I wanted to adopt it fifty years ago, but it seemed then that it was just not used, so I stayed with Euclid). No amount of mathematical "proof" can change the meaning of words. We can say "It would be more logical if ...", but if the words are not used in the way we would like, then we just have to accept reality. Dbfirs 12:44, 13 November 2009 (UTC)

Ahh, the case where one miscomprehends defn #2. Well, rather than repeating the points I made earlier that you left out, I will simply agree with your statement that it will not help to clarify the article to include all four. If one (mis)interprets defn #2 in that manner, they will still see it explained in defn #1. However, I would like to comment on your statements in the latter half of the paragraph. There is a certain reality that your position does not take into account. We accept that a polygon falls into the same classification if it has or inherits the same properties. Without going into detail, we can say a square inherits the same properties as a rhombus and a rectangle. Therefore a square is a rhombus and a rectangle. Same thing true regarding a rhombus and rectangle to a parallelogram. This inheritance of properties refutes Euclid's claim regarding separate classifications for these shapes. Applying the exact same methodolgy, a similar situation exists for equilateral and isosceles triangles. Since an equilateral triangle inherits all the properties of an isosceles triangle, it is an isosceles triangle. This once more refutes Euclid's noninclusive classifications. The inheritance of properties is the underlying foundation for the various classifications of shapes. Regardless of the interpretations of the definitions, the shape's properties cannot simply be ignored.

Having said all that, three definitions still remain. Do you have a suggestion on how to present them?

Also, I hit the size warning once again. I'd like to archive (actually delete) much of our earlier isosceles discussion since it consumes much of this page. Sound like a plan? JackOL31 (talk) 19:30, 14 November 2009 (UTC)

We are just not getting anywhere are we? Each of us thinks that the other has misunderstood the definition. Why don't we just leave things as they are (with one definition and a note), archive the ridiculously lengthy discussion (why have we wasted so much time on such a trivial issue?), and spend our time on more productive editing? Dbfirs 21:24, 14 November 2009 (UTC)
I did not notice before that the definition was changed back. I'm pleased with the way it is now. However, the note is biased. I know mathematicians that have never agreed with Euclid's definition. Learned it, yes. Followed it, no. It needs a NPoV, such as some mathematicians agree with Euclid's definition excluding equilaterals, while some mathematicians allow equilaterals as isosceles. There is no tradition here, either you agree with it or you don't. I do agree with the archiving (I say all of it) and back to productive editing. Sadly, productive editing appears to be undoing vandalism. JackOL31 (talk) 22:41, 14 November 2009 (UTC)
I'll make the change to the wording of footnote #2 after the holidays when I have more time. JackOL31 (talk) 00:45, 18 November 2009 (UTC)
The following is my proposed rewording for Reference #2: Euclid defines isosceles triangles based on the number of equal sides, i.e. only two equal sides. An alternative approach defines isosceles triangles based on shared properties, i.e. equilateral triangles are a special case of isosceles triangles.
The proposed update changes the reference to a matter-of-fact tone. It also removes the unverifiable statement (opinion) what mathematicians follow or do not follow. I would also propose keeping the links. JackOL31 (talk) 17:55, 21 November 2009 (UTC)
That sounds a very fair compromise, and an improvement on the old note. I'm happy to support your change. Dbfirs 17:16, 23 November 2009 (UTC)

### Requested outside comment

(Full disclosure: JackOL31 approached me to help mediate this dispute, presumably because I put a welcoming message on their talk page.)

Mathematically speaking, I think the definition of isosceles is completely self-evident and that only an idiot wouldn't agree with my personal opinion on which one is correct. Good thing it doesn't matter, editorially speaking, what anyone here thinks. :-) It appears from what's posted here that there are two definitions of isosceles floating around and that neither one is more prominent than the other, although surely many trees have sacrificed themselves for the cause and mountain of text could be found arguing either way. I suggest that the definition be worded something like this:

"An isosceles triangle can be defined either as having exactly two equal sides or as having at least two equal sides."

The italics should remain in the text to emphasize the difference for lay readers and the backing source should be after each definitional phrase, rather than at the end of the sentence together, to avoid confusion. It might be interesting to add a note about why the definition is important and contentious or a historical note, but is probably not necessary.--Gimme danger (talk) 04:55, 11 August 2009 (UTC)

Thanks for your mediation. I'm happy with your decision. Dbfirs 15:00, 11 August 2009 (UTC)
The concern with that approach is that it is not consistent with what is published for parallelograms. That page does not use the EE (Euclid Exclusionary) definitions. Instead, it uses the properties of the set approach. Thus, we have a square is a rhombus and a rectangle, and a rhombus and a rectangle are a parallelogram. To be consistent, shouldn't we use the dual approach with parallelograms and its other types or do we use different rules for different sets of polygons? It also begs the question why the EE definitions aren't used for quadrilaterals. Why, because it doesn't work when a parallelogram, rhombus, rectangle and square are all mutually disjoint. This begs another question, if EE definitions don't work well with quadrilaterals, what leads us to believe it will work any better for triangles?
Let's also keep in mind the bent straw example. Bend a straw in half, hold it upright on a table, and slowly separate the ends until it is horizontal on the table. This will generate 3 subsets: acute isosceles triangles (0<vertex<90), right isosceles triangles (vertex=90) and obtuse isosceles triangles (90<vertex<180). Now, although you're generating all possible triangles from the same two equal sides, you decide that equiangular is just too different to be a member of acute isosceles. So you remove it and make it a sibling set. Wait, it can't be a fourth subset, it has to be removed one more time (since it's equilateral) and become a sibling to the parent of the set it was removed from! Of course, straight up that's just mathematical nonsense. To put this in perspective, as an example, it is equivalent to saying that as you fix a set of parallel sides and push in a pull out these sides to form all the possible rectangles, you decide that the square just isn't a rectangle and you pull it out. But now, you also say it is not a parallelogram either and pull it out to be a sibling set to parallelogram. How can something generated within a subset not even be a member of the parent set? This is a monumental mathematical contradiction and it only takes one contradiction to prove something false. Actually, there is a second contradiction. Although you pulled squares out of rectangles, it has to share some properties with rectangles to be generated from them. Why wouldn't those shared properties keep it under its parallelogram parentage? Better yet, since they shared some properties, why weren't rectangles pulled out, too? Of course, you can't pull out squares from parallelograms and leave in rectangles. Similarly, you can't pull out equiangular and leave in acute isosceles (except that you are doing just that). Bottom line, if EE definitions do not work for quadrilaterals, how can we expect them to work for triangles? C (talk) 03:05, 12 August 2009 (UTC)
It seems to me like you're missing a key point of Wikipedia policy/philosophy. It's okay; you're new and there's a lot of exciting policy to learn, most of which is far more entertaining than watching paint dry. On a fundamental level, we are not here at Wikipedia to decide what is true or not, but rather to report what others have said about things. "Verifiability, not truth" is the catchy phrase. If there are alternate definitions for parallelograms in circulation, then those definitions should all be present on the parallelogram page. If there are alternate definitions for isosceles, those definitions need to be present. Wikipedia itself can't really take a stand on anything. Does that make sense?
I understand the impulse to want the right answer on this page. As I said, I have a strong opinion on the subject myself. But that's not really the purpose of Wikipedia, I guess. --Gimme danger (talk) 03:27, 12 August 2009 (UTC)
... and JackOL31 was happy to accept EE definitions for diagonals of quadrilaterals and pentagons. I think the argument is more one of semantics than of mathematics. Dbfirs 12:04, 12 August 2009 (UTC)
I'm afraid you misunderstand the mathematics, I can say with certainty that I never accepted any EE definitions in our discussion. Two diagonals and 4 sides make a 1-to-1 relationship, they are therefore the same set. I have noticed that you have made several errors in this area of mathematics, so I think it best not to assume an error of semantic misunderstanding on my part. JackOL31 (talk) 03:47, 13 August 2009 (UTC)

For Gimme danger - as long as you understand my point that one page says that Mars was created by accretion while the other states that the Earth was spun off from the sun, if you can follow my rough allusion. JackOL31 (talk) 04:04, 13 August 2009 (UTC)

... but don't you see the contradiction? The set of polygons with two diagonals cannot be the same set as the set of quadrilaterals, because any pentagon has two diagonals, but a pentagon is not a quadrilateral. It all depends on whether "two" means at least two or exactly two. Dbfirs 21:19, 13 August 2009 (UTC)
...In all seriousness, I can't. The first mathematical fallacy in your argument is that a pentagon has two diagonals, when it actually has five diagonals. The number of sides, the number of angles and the number of diagonals for polygons are by exact count! A triangle does NOT have one angle. The set of all polygons having 2 diagonals is the set of quadrilaterals. The set of all polygons having "at least" 2 diagonals is a collection of sets enumerated by the set of Quadrilaterals UNION Pentagons UNION Hexagons UNION..., or more simply put, Polygons \ Triangles (set subtraction). However, the number of "equal" sides or "equal" angles can subsume a larger number of "equal" sides or "equal" angles. In other words, the property of equality can vary and could possibly be subsumed, but you have to let the chips fall where they may. For triangles, a triangle with 0 equal sides cannot have 2 equal sides and vice versa, but a triangle with 2 equal sides can have 3 equal sides and a triangle with 3 equal sides must have 2 equal sides (a clue as to which is a subset of the other). So, if you don't get caught up in subsuming things that have exact counts, you must then avoid the second mathematical fallacy of subsuming in the wrong direction. Two sets of equal sides (not "exactly" two) subsumes four equal sides, not four equal sides subsumes two sets of equal sides. The key to defining subsets is to define enough properties to make it disjoint from the others, but not overly defined so as to preclude ITS subsets. Unfortunately, Euclid does preclude the subsets of parallelograms, in contrast to the Wiki page on parallelograms which does not use the EE definitions. Now, if we avoid the EE definitions for triangles, we see a subset for isosceles triangles, namely equilateral triangles. But sadly, for triangles we are primarily following the EE definitions. Why? Solely because Euclid defined them that way 2300 years ago. Of course, this is the similar way Euclid defined quadrilaterals 2300 years ago, but definitions we've rejected because counting the number of equal sides didn't really work out - it made all subsets of parallelograms disjoint from each other and itself. But we believe this failed method will work much better on a different set of polygons, the triangles. And the mathematical reason for this is...? And did we notice that the EE isosceles definition made equilateral triangles disjoint from itself? So we must be saying that this is all a coincidence, not anything mathematical going on here, right? JackOL31 (talk) 23:32, 15 August 2009 (UTC)
... so what about the set of polygons with two equal diagonals? Dbfirs 07:52, 16 August 2009 (UTC)
... so how about acknowledging my points in the previous posting first? Please either agree to their validity or try to refute them. You can skip the point you're trying to refute above and address the others. JackOL31 (talk) 13:22, 16 August 2009 (UTC)
... Of course I acknowledge the validity of your argument, but it is not the only argument, and it, also, can lead to contradictions. Dbfirs 14:51, 16 August 2009 (UTC)

I'm glad you acknowledged my statements, sometimes I feel as if I'm talking to the wall. One can always make another argument, but that doesn't mean it will stand up mathematically. Anyway, we'll cover that later. Regarding contradictions, that has yet to be shown. Regarding your question, you'll have to tell me what you think is the set of polygons with two equal diagonals. Then, tell me where you believe the contradiction lies and I'll go from there. Bear in mind that you may create a subset that has no additional shared properties, especially when you pull them from disjoint sets. For example, the subset right triangles comes from scalene and equilateral triangles. They don't share anything else in common other than having right triangle and triangle properties. JackOL31 (talk) 20:13, 16 August 2009 (UTC)

No, you are not talking to the wall, but neither are you talking to one of George Orwell's sheep on the subject of two legs. May I ask you to clarify your statement on when two must always mean just two, and when it means at least two? Dbfirs 22:23, 16 August 2009 (UTC)
I hope you understand that "the wall" means that it is as if no one is there when I explain, since I get very little response. Not sure of the intent of the Animal Farm reference is, but I'll move on to the clarification. As I understand it when discussing polygons, if it has one fixed value then that means exactly. As mentioned earlier, a quadrilateral has four sides. So, polygons with four sides are quadrilaterals. The number of sides, angles and diagonals are unconditional. If you want to override the implied exactly, then something along the lines of, "polygons with at least 5 sides" or "polygons with at most 10 sides" will suffice. However, there are no fixed numbers of "equal" anything associated with general polygons. Therefore, if you wanted the set of polygons with two equal diagonals, then any two equal diagonals will do (yes, more "any" than "at least"). As soon as you see two equal diagonals, the condition has been met. Of course, you could always restrict it by saying, "quadrilaterals with two equal diagonals (rectangle)" or "a polygon with exactly two equal diagonals and only one pair of equal diagonals". I suppose it can get quite complex, so you may need to add verbage to uniquely identify what you're interested in collecting. Put in whatever words are necessary. I've found that these sets aren't really discussed much since they don't generate anything interesting. Usually, one breaks down a particular polygon into its subsets. Reaching across disjoint sets doesn't lead to anything useful (IMHO). How closely related can an equilateral triangle and a 74-gon with three equal sides be? JackOL31 (talk) 00:31, 18 August 2009 (UTC)
Are we not correct, then, if we "put in whatever words are necessary" when defining isosceles triangles?
The sheep in Animal Farm changed their chant from "four legs good; two legs bad" to "four legs good; two legs better" without noticing that they had changed the meaning. I don't think they discussed whether the legs were equal. Dbfirs 07:14, 19 August 2009 (UTC)
What I was referrring to was the use of necessary words to precisely identify the objects of interest. This has nothing to do with how to name them. Regardless of the semantics, the mathematical concept remains the same. There is a set of triangles with two equal sides (regardless of the base length) that share the same properties (illustrated by the bent straw). If any two triangles formed from the two equal sides share a set of properties, then all triangles formed from the same two equal sides share the same set of properties. Equilateral triangles have additional properties not found in the other triangles from the set. This make equilateral triangles a subset of the larger set. Now, to call the entire set isosceles is consistent with the naming conventions we use for parallelograms, rhombuses and rectangles (and others). The name of the set is the name of the shape that contains the subsets. For example, a rhombus does not mean non-square rhombuses. Another significant consequence of this larger set is that it contains all the members that are not scalene. This set IS the sibling to scalene. Therefore, we can reject any definition that does not use this set as the sibling set to scalene.
Thanks for sharing those passages from Animal Farm. By the way, adding the words "exactly" or "only" changes the meaning of the definition. I have consistently not used those words. However, now I feel I must reciprocate by sharing some other literary passages. I choose Flatland, written by British mathematician Edwin Abbott Abbott in 1884. The passages read, "...Rarely - in proportion to the vast numbers of Isosceles births - is a genuine and certifiable Equal-Sided Triangle produced from Isosceles parents. ... The birth of a True Equilateral Triangle from Isosceles parents is the subject of rejoicing in our country for many furlongs round." JackOL31 (talk) 13:18, 23 August 2009 (UTC)
Thanks for sharing Abbott's satire, but didn't the achievement of equal angles remove these offspring from the class? Dbfirs 16:47, 23 August 2009 (UTC)
Yes, and the next class up is Squares and Pentagons. I don't have anything more to share, so I am going to assume this discussion has reached its conclusion. JackOL31 (talk) 00:26, 25 August 2009 (UTC)
... which meant, of course, that they joined the class of regular polygons, and were no longer classed as isosceles, but I agree that we are not getting anywhere. Dbfirs 09:17, 9 November 2009 (UTC)
... they were removed so they wouldn't devolve back into general isosceles. The other stmt refers to the fact that Abbott's classes become a mixture of sets and are therefore meaningless in our discussion. I did enjoy the story. JackOL31 (talk) 01:50, 13 November 2009 (UTC)
... Agree 100% here! Dbfirs 21:24, 14 November 2009 (UTC)

I am only going to add a comment here because the dispute involved highlights one aspect of how geometry is taught in the US, at least. In the textbook used by the district in which I teach, isosceles triangles are taught as having "at least" two congruent sides (PLEASE don't get excited about congruent v equal!). However, trapezoids are taught as having ONLY one set of parallel sides (thus, parallelograms are not a subset of trapezoids), and kites are taught as excluding rhombuses, for no apparent reason. Thus, the authors of the text do not have a co-ordinated approach to the concept of when to include one classification as a subset of another, and when to leave the classifications disjoint.

Quaere: does anyone in mathematics discuss this outside of textbooks? That is, are there any articles discussing it? When and why did the definition included in modern American geometry texts ("at least") start being used for the isosceles triangle? Doug (talk) 20:02, 1 May 2010 (UTC)

Unfortunately, I can't answer your questions. I've tried and failed to find the answers. The inconsistencies have puzzled me, too. Nearly fifty years ago, I asked why equilateral triangles were not also isosceles, and was told that Euclid divided triangles into three disjoint subsets (or words to that effect). By contrast, all definitions for quadrilaterals in the UK are inclusive. I'm puzzled by the discrepancies, but I think we just have to accept that they exist. I suppose that we all have the (religious) tendency to believe that what we were first taught is the only acceptable truth. I suspect that, eventually, in a century or two, only the inclusive definitions will survive, but text-book writers tend to reproduce what they were taught, so traditions are preserved well beyond their useful life. Another cause of confusion is the usage of "trapezium" and "trapezoid", though I think that they will eventually become synonyms. Dbfirs 08:29, 3 May 2010 (UTC)