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Archive 1 | Archive 2 |
Oh, dear! The diagrams in this page are GOOD! Whoever did them did a good job! Pfortuny 21:49, 31 Mar 2004 (UTC)
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?
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)
The information on this article is a bit disjointed. Where are the references? - Ta bu shi da yu 13:12, 19 Dec 2004 (UTC)
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)
∆ links here, but isn't that the greek letter Delta (letter)? ���� 213.112.14.187 07:54, 8 March 2006 (UTC)
"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)
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).
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.
The formula
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
A proof for this can be found at http://mcraefamily.com/MathHelp/GeometryTriangleAreaVector2.htm -- anonymous
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)
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)
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.
This is cute, but excessive!
Why not just use difference vectors and a cross product: A=(Ax,Ay,Az), B=(Bx,By,Bz), C=(Cx,Cy,Cz)
Tom Ruen 03:24, 22 November 2006 (UTC)
Please add a section dealing with equal triangles. The Ubik 18:08, 4 December 2006 (UTC)
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 and and also the formula shown above, then one arrives at the following formula for area [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.) Using the property of the sine curve, namely sin X = sin (180-X) one arrives at three more formulae |
is not necessary because
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
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
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.
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
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.
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)
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)
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)
(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)
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)
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)
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 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)
"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)
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)
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
A=.5ab*sine of included angle —Preceding unsigned comment added by 66.65.139.242 ( talk) 07:52, 19 January 2008 (UTC)
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)
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)
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)
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)
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)
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)
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)
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 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'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
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)
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)
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)
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)
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)
Would someone please tell readers what program was used to draw the diagrams and write the equations, they are very well done.
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)
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.
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)
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)
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)
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)
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.
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)
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):
www_askoxford_com/concise_oed/isosceles?view=uk
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)
Definitions were clear in the past, before modern "Algebra of Sets" mathematicians started tinkering! Dbfirs 10:38, 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)
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)
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)
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)
(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)
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)
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)
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)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
Oh, dear! The diagrams in this page are GOOD! Whoever did them did a good job! Pfortuny 21:49, 31 Mar 2004 (UTC)
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?
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)
The information on this article is a bit disjointed. Where are the references? - Ta bu shi da yu 13:12, 19 Dec 2004 (UTC)
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)
∆ links here, but isn't that the greek letter Delta (letter)? ���� 213.112.14.187 07:54, 8 March 2006 (UTC)
"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)
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).
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.
The formula
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
A proof for this can be found at http://mcraefamily.com/MathHelp/GeometryTriangleAreaVector2.htm -- anonymous
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)
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)
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.
This is cute, but excessive!
Why not just use difference vectors and a cross product: A=(Ax,Ay,Az), B=(Bx,By,Bz), C=(Cx,Cy,Cz)
Tom Ruen 03:24, 22 November 2006 (UTC)
Please add a section dealing with equal triangles. The Ubik 18:08, 4 December 2006 (UTC)
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 and and also the formula shown above, then one arrives at the following formula for area [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.) Using the property of the sine curve, namely sin X = sin (180-X) one arrives at three more formulae |
is not necessary because
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
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
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.
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
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.
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)
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)
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)
(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)
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)
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)
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 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)
"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)
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)
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
A=.5ab*sine of included angle —Preceding unsigned comment added by 66.65.139.242 ( talk) 07:52, 19 January 2008 (UTC)
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)
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)
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)
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)
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)
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)
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)
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 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'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
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)
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)
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)
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)
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)
Would someone please tell readers what program was used to draw the diagrams and write the equations, they are very well done.
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)
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.
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)
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)
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)
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)
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.
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)
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):
www_askoxford_com/concise_oed/isosceles?view=uk
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)
Definitions were clear in the past, before modern "Algebra of Sets" mathematicians started tinkering! Dbfirs 10:38, 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)
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)
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)
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)
(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)
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)
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)
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)