Talk:Triangle

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Triangle is a former featured article. Please see the links under Article milestones below for its original nomination page (for older articles, check the nomination archive) and why it was removed.

This article appeared on Wikipedia's Main Page as Today's featured article on May 14, 2004.

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May 5, 2004

Featured article candidate

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February 25, 2005

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January 4, 2006

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January 17, 2006

Good article reassessment

Delisted

Current Status: Former featured article

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Needs references, and also better lead, and more introduction. Geometry guy 23:08, 9 June 2007 (UTC)

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	The following comments have been left for this page: Needs references, and also better lead, and more introduction. Geometry guy 23:08, 9 June 2007 (UTC) (edit)

1 Euclidean Geometry
2 Major browser compatibility problems with article
3 Talk
4 Congratulations
5 Questions
6 Excellent work... however
7 Sum of the angles of TRIangle
8 Just a thought
9 Equilateral Triangles - Another Way to Calculate Area
10 Removed text from lead
11 ∆
12 Equilateral vs Equiangular
13 Equilateral triangle existence
14 Wrong formula for the area of the triangle
15 Proof that angles in a triangle sum to 180 degrees
16 Monster formula
17 Equal Triangles ?
18 Triangles
19 New formula for area of a triangle
20 What should be included
21 Simplest Area Formula
22 right triangle
23 solving a triangle
24 Centroid/barycenter confusion
25 Congruence
26 Error in a definition
27 Another error in a definition (isosceles triangle)
28 Isn't there another formula?
29 About the sentence...
30 scalene triangle.
31 Graphing a Triangle on a Cartesian coordinate system
32 Another Triangle Formula

[edit] 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)

[edit] Major browser compatibility problems with article

This article is filled with content which is not rendering properly in mainstream browser configurations.

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).

2. There are links for missing (.svg) images in the "Types of Triangle" section

3. Code like this: ":[math]c^2 = a^2 + b^2 \,[/math]" (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 ":[math]...[/math]" 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)

[edit] 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)

[edit] 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)

[edit] Questions

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

[edit] Excellent work... however

The information on this article is a bit disjointed. Where are the references? - Ta bu shi da yu 13:12, 19 Dec 2004 (UTC)

[edit] 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)

[edit] 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)

[edit] 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 ${\sqrt{3} \over 4} s^2$ using <math>{\sqrt{3} \over 4} s^2</math> --Henrygb 10:09, 30 October 2006 (UTC)

[edit] Removed text from lead

I removed this recent addition to the lead section:

"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)

[edit] ∆

∆ 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)

[edit] 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)

[edit] 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.

[edit] Wrong formula for the area of the triangle

The formula

$S=\frac{1}{2}\sqrt{\begin{vmatrix}x_1 & x_2 & x_3\\ y_1 & y_2 & y_3 \\ 1 & 1 & 1\end{vmatrix}^2+ \begin{vmatrix}y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \\ 1 & 1 & 1\end{vmatrix}^2+ \begin{vmatrix}z_1 & z_2 & z_3\\ x_1 & x_2 & x_3 \\ 1 & 1 & 1\end{vmatrix}^2}$

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

$S=\frac{1}{2}\left(\begin{vmatrix}x_1 & x_2 & x_3\\ y_1 & y_2 & y_3 \\ 1 & 1 & 1\end{vmatrix}+ \begin{vmatrix}y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \\ 1 & 1 & 1\end{vmatrix}+ \begin{vmatrix}z_1 & z_2 & z_3\\ x_1 & x_2 & x_3 \\ 1 & 1 & 1\end{vmatrix}\right)$

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:

$S=\frac{1}{2}\sqrt{\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1\end{vmatrix}^2+ \begin{vmatrix} 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{vmatrix}^2+ \begin{vmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 1\end{vmatrix}^2} = \frac12 \sqrt{1^2+0^2+0^2} = \frac12.$

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

$S = \frac12 \big( |u \times v| + |v \times w| + |w \times u| \big),$

which is not the same as

$S = \frac12 \big| u \times v + v \times w + w \times u \big|.$

-- Jitse Niesen (talk) 09:30, 30 June 2006 (UTC)

[edit] 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)

[edit] Monster formula

This is cute, but excessive!

$S=\frac{1}{2} \sqrt{ \left( \det\begin{pmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} y_A & y_B & y_C \\ z_A & z_B & z_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} z_A & z_B & z_C \\ x_A & x_B & x_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 }.$

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)

[edit] Equal Triangles ?

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

[edit] Triangles

I love triangles. —The preceding unsigned comment was added by 205.219.133.241 (talk) 02:33, 6 December 2006 (UTC).

[edit] 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.

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

[edit] 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.

If one uses $\cos C=\frac{a^2+b^2-c^2}{2ab}$ and $\sin C = \sqrt{1- \cos^2 C}$ and also the formula shown above, then one arrives at the following formula for area $\frac{1}{4} \sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}$ [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.) $\frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ca\sin B$ Using the property of the sine curve, namely sin X = sin (180-X) one arrives at three more formulae $\frac{1}{2}ab\sin (A+B) = \frac{1}{2}bc\sin (B+C) = \frac{1}{2}ca\sin (C+A)$

is not necessary because

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.
The formulas

$\frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ca\sin B$

and

$\frac{1}{2}ab\sin (A+B) = \frac{1}{2}bc\sin (B+C) = \frac{1}{2}ca\sin (C+A)$

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.

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.
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. --KSmrq^T 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

sin A

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 S =½ab sin C and S=½bc sin A</math> are actually different formula from S=½ab 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. --KSmrq^T 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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:

${6 \pm\sqrt{32} \over 2}$

Reduce these to:

${3 \pm\sqrt{8} \over 1}$

(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:

${P \pm\ R*\sqrt{D} \over Q}$

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

P 2 - R 2 * D

In our example, we reduce further:

${3 \pm\ 2*\sqrt{2} \over 1}$

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).]

[edit] 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)

[edit] About the sentence...

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)

[edit] scalene triangle.

Every one help the planet and save our earth keep it clean. —Preceding unsigned comment added by 58.167.199.47 (talk) 10:39, 26 March 2008 (UTC)

[edit] 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 (talk • contribs) 14:33, 16 April 2008 (UTC)

[edit] 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)