Talk:Triangle/Archive 2

Minor change re: isosceles triangle definition
I've slightly modified the sentence about two definitions for isosceles triangles (and I'm glad to note that the dispute on that got ironed out so even-handedly). The old wording said that "some definitions state that ..." and mentioned the definition of an isosceles triangle as having only two congruent sides. This is imprecise. There is only one definition that has this quality, whereas "some definitions" implies that there are a number of them out there, each with this particular sub-point included. So I've changed it to state that some mathematicians define them to have only two congruent sides, whereas others use the "at least two" concept.

Hopefully, this doesn't cause another full scale battle to erupt. :) Doug (talk) 19:43, 1 May 2010 (UTC)


 * It was intended to imply "some sources", but you have a valid point. The sentence could be still better expressed, but perhaps this should be done by an independent editor for the reasons you mention!    D b f i r s   08:03, 3 May 2010 (UTC)

Barycenter inappropriate terminology for triangles
User:Woodstone, the continued effort to add the term "barycenter" as an alternative for "centroid" is misplaced. A barycenter, by definition, is a three-dimensional concept. A triangle is two-dimensional; it has no barycenter. A trianglular object, of uniform thickness and density will have its barycenter at a point which corresponds to the centroid of the triangular face of the object, but that fact is documented later in the paragraph on the centroid. If you wish to note that the balancing point discussed there is called a barycenter, feel free. Doug (talk) 21:26, 24 May 2010 (UTC)


 * The repeated effort to suppress mention of the term barycenter is misplaced. It is is a term that is in actual use regarding triangles. That you are of the opinion that it is a loose usage of the term is not conclusive. Wikipedia reports on actual use, not on an idealised reflection. I have given a reference to support this. &minus;Woodstone (talk) 21:55, 24 May 2010 (UTC)


 * Your reference does not support the statement that a "centroid" of a triangle (the point of concurrence of the medians) is "the barycenter of the triangle. A barycenter, as your referenced article notes, is based upon the mass of the points in question.  IF you have three masses at the vertices of a triangle, and IF the masses are equal, THEN the centroid of the triangle formed by considering the masses as points in a plane would coincide with the barycenter of the triple masses.  This is NOT the same thing as your assertion that the centroid discussed in the article is also known as the barycenter.


 * Unless and until you can provide support for the statement that this is a common, or even uncommon, but regularly used by a significant portion of the mathematical world, term for the point commonly referred to as the centroid of a triangle, please do not return this statement to the article. Doug (talk) 01:48, 25 May 2010 (UTC)


 * There is no conspiracy to suppress the term barycentre, though I have never heard it used for triangles. The usual meaning of barycentre is for the centre of mass of a sun-planet or a planet-moon system.  Centroid (meaning centre of area or centre of volume) is a precisely defined term understood by most (I would have thought all) mathematicians.  The centroid coincides with the centre of mass for a uniform mass distribution, and this coincides with the centre of gravity in a uniform gravitational field, but the terms are not synonyms, they just happen to coincide under certain conditions.    D b f i r s   06:36, 25 May 2010 (UTC)


 * We have an article on barycentric coordinates, and the term comes from "barycentre", which is a synonym for "centre of mass". The centre of mass can be calculated in any number of dimensions - it is not specifically a three-dimensional concept. However, I cannot find any support for using "barycentre" as a synonym for "centroid" in a triangle. Gandalf61 (talk) 07:28, 25 May 2010 (UTC)


 * You should have looked to the article Barycentric coordinates (mathematics). The use of barycentric coordinates devoid of any notion of mass is common in computer graphics and finite element analysis. For that reason alone, it is worthwhile to identify the term barycenter in the article. In the context of mathematical triangles, the "barycenter" is equivalent to the centroid. It is the average of all interior points of the triangle in a vector space. A possible explanation in terms of masses on the corners may sometimes be given for historical reasons, but is not necessary in any way. See the following references:
 * math.uky.edu
 * utdallas.edu
 * devmaster.net
 * paideiaschool.org
 * farinhansford.com
 * You may also note that of the transwiki links in centroid, many refer to a name that is a cognate of barycenter.
 * &minus;Woodstone (talk) 12:15, 25 May 2010 (UTC)


 * Yes, barycentric coordinates (mathematics) was the article that I meant to link to. The only place that it uses the word "barycenter", it uses it as a synonym for centre of mass. Some of the other references on your list may support your contention that barycentre is also a synonym for centroid, but published books or journals are more likely to be accepted as reliable sources. Gandalf61 (talk) 13:09, 25 May 2010 (UTC)
 * The exact word barycenter with reference to mass points in the article comes as an afterthought in the historical paragraph. The word barycentric appears throughout without reference to mass. You are twisting evidence here. For more references you may try this. &minus;Woodstone (talk) 14:15, 25 May 2010 (UTC)


 * Hostility and rudeness towards those who are trying to assist you will not advance your position. I am done here. Gandalf61 (talk) 14:28, 25 May 2010 (UTC)


 * I'm sorry if my criticism on your interpretation offended you. But walking away is not doing any good. Please have a look at the abundant evidence in the book search. &minus;Woodstone (talk) 15:59, 25 May 2010 (UTC)


 * There is common confusion between the various centres, and, perhaps in some contexts, barycentre is an appropriate term, but why must a triangle necessarily have mass (or uniform density)? The fundamental definition of a triangle is a 2-D shape, so mass is impossible unless you allow infinite density.    D b f i r s   16:53, 25 May 2010 (UTC)


 * The question is not "is this logical" from a certain viewpoint, but "is it used this way" in the real world. The original name may be derived from a weight concept, but in modern theory it is just used as average coordinate (also called center of area). There is no need to assume any presence or distribution of mass. &minus;Woodstone (talk) 17:01, 25 May 2010 (UTC)


 * No, the average coordinate (also called center of area) is the centroid, not the barycentre. One of your own references directly refutes your claim by specifying the conditions under which the barycentre coincides with the centroid.    D b f i r s   17:07, 25 May 2010 (UTC)


 * Now it's getting very confusing. Earlier on you said a triangle cannot have a barycentre. Now you are saying the barycentre is not the centroid? So there is a barycentre. How would you define it? &minus;Woodstone (talk) 21:03, 25 May 2010 (UTC)


 * Yes, of course a triangle can have a barycentre if it has mass or some other weighting. The definition of barycentre essentially includes the weighting in most texts, though others define it as just the centre of mass.  An "ideal" triangle (per Euclid) would have no thickness and thus no mass, but many texts use "triangle" to mean a triangular lamina.  It was one of the texts that you linked to that said the barycentre is not always the centroid.    D b f i r s   00:34, 26 May 2010 (UTC)


 * I'd be happy with adding something along the lines of "... centroid, also called barycentre (barycenter) for a uniform triangular lamina".   D b f i r s   14:29, 27 May 2010 (UTC)


 * The article "centroid" links to 17 other language wikis. Of those words, 4 I cannot decipher, 8 are cognates of centroid, and 5 of barycenter. So at least in those 30% languages (including Greek) it is standard to use such a word in the context of pure euclidian triangles. In the application of barycentric coordinates the concept of mass is normally not present and no assumption of a 3rd dimension is implied. In computer graphics they are used to decide if a line in space intersects a bounded triangle. This is just pure geometry. In finite element analysis, the barycentric coordinates are used to define a basis for the piecewise linear function space on the trilateral mesh. These are just pure scalar functions. The basis is for example used to define a field of temperatures, pressures, charges, or indeed possibly densities, but the latter is certainly not the most common. So I still would prefer to mention the term "barycenter" as an alternative name&mdash;in the context of triangles&mdash;of the centroid. I would not recommend mentioning a laminate in a geometry article. &minus;Woodstone (talk) 16:12, 27 May 2010 (UTC)


 * You are, I believe, confusing issues. The fact that barycentric coordinates are used in two-dimensional analysis does not mean that the point of concurrence of the medians of a triangle is called the barycenter by anyone as a regular term.  It's only true when the coordinates of the vertices of the triangle are equally weighted (such as by (1,0,0), (0,1,0) and (0,0,1), but the barycenter of a triangle with vertices unequally weighted would not be at the point of concurrence of the medians.  You have yet to provide documentation that the term "barycenter", which has a specific meaning (center of a system of masses), is used to discuss the point of concurrence of the medians of a triangle, even in cases where people are using barycentric coordinates in the plane.  And, as has been pointed out more than once in this discussion, this may well be because the barycenter does not need to coincide with the centroid.  The fact that some small branch of analysts may use the terms interchageably in the limited instances they work with does not mean Wiki should be amended to reflect this as some general usage of the term barycenter. Doug (talk) 20:31, 27 May 2010 (UTC)


 * Have you had a look at the book search given above? Most of the book excepts in the first page of results clearly define the barycenter as intersection of the medians or as the vector (A+B+C)/3. Computer graphics and finite elements are certainly not some small vague branches of analysis, but rather extensively used applications of geometry. I think you are still focusing too much on the astronomical use of the word barycenter. And actually, that is not defined for triangles, but for several (perhaps three) masses somewhere in space. In short: I am not confused and have plenty of references for the use of barycenter as identical to centroid in the context of triangles. &minus;Woodstone (talk) 21:34, 27 May 2010 (UTC)


 * I have looked at them. Probably better than you have, apparently.  Enough to recognize that several of them in the brief limited look available make mention of exactly the point I was making.  For example, look at the reference in the book on Linear Algebra, where it notes that the barycenter conincides with the centroid only when the coordinates of the vertices are equally weighted.  The bottom line remains:  the barycenter and the centroid are not the same thing.  They may coincide, but they do not have to coincide.  Therefore, it's inappropriate to assert that the centroid of a triangle is automatically its barycenter.Doug (talk) 01:52, 29 May 2010 (UTC)


 * The word barycentre derives from the Greek βάρος, meaning weight, so it is normally interpreted as meaning centre of mass. In the case of a uniform triangular lamina, the term barycentre can be used as a synonym of centroid because it is easy to prove that they coincide.  In other circumstances, they will not coincide.  At least one of the texts that you keep mentioning specifically distinguishes between the terms.    D b f i r s   22:08, 27 May 2010 (UTC)

So why is the Greek version of the article centroid given as Κέντρο βάρους? Evidently the Greeks do not object to the use of βάρος for the centroid? This article is not about laminates, but about triangles. They have no mass. So there can be no confusion when using this term in context of triangles, as is done in many text books as cited. &minus;Woodstone (talk) 22:29, 27 May 2010 (UTC)
 * ... except for the ones that say they are different! ... and the article Κέντρο βάρους is about centre of mass (as you would expect)    D b f i r s   22:32, 27 May 2010 (UTC)
 * I would support adding a mention of barycentric coordinates to the article, though I would claim that they do not necessarily have their origin at the centroid.   D b f i r s   13:55, 1 June 2010 (UTC)


 * Barycentric coordinates do not really have an origin, since there is the requirement that the sum of the coordinates is 1. The centroid is at (1/3,1/3,1/3). A term that I saw in several of the references and that may be acceptable to the you is "geometric barycentre". It expresses that the geometry of the triangle itself is weighted, not some attached masses. By the way, I know WP is not considered a reliable source, but Baricentro (geometria) and Zwaartepunt (zwaar = heavy) are clear about their meaning. &minus;Woodstone (talk) 17:18, 1 June 2010 (UTC)
 * Yes, I would be happy with "geometric barycentre" since the implied weighting would not be by mass. Thanks for the links to other Wikipedias - the usage does seem to vary considerably.  Do you not regard "centroid" as clear and unambiguous?    D b f i r s   21:59, 1 June 2010 (UTC)
 * In my personal perception "centroid" is a vague term. Compare "humanoid". Generally a word like "xxxoid" means "similar to an xxx", and it encompasses a whole group of entities. So the word itself is nondescript. Evidently it has been given a more specific meaning in some mathematical circles. Please note that I have never proposed to remove the word "centroid", just to add barycenter in some form. I still do not understand where the fierce opposition to barycenter comes from. In geometric context this word is clear to me, as in a Euclidian (2D) triangle, the only thing that can be "weighted" is the geometry itself. So I'm glad we seem to have a compomise in making that explicit in the term "geometric barycenter". &minus;Woodstone (talk) 08:10, 2 June 2010 (UTC)
 * I have the opposite perception that centroid is a precisely defined term, and that barycentre is used in various differing ways depending on context (with some implied weighting implicit in the usage), but I agree that specifying geometric barycentre makes it unambiguous and a useful addition to the article.   D b f i r s   08:43, 2 June 2010 (UTC)

Misleading incircle diagram
I think that in the section "Points, lines and circles associated with a triangle", the graph labelled "The intersection of the angle bisectors is the center of the incircle" is unintentionally misleading. Unless you look at it very carefully, it looks like the intersection of any angle bisector with the opposite side of the triangle occurs at the incircle tangency. (For two of them it looks exact, and the other one looks very close.) But this is not in general true. Could someone redraw the graph to make this clear? Duoduoduo (talk) 16:44, 12 June 2010 (UTC)
 * None of them looks exact on my laptop, but I agree that only one bisector crosses at a point clearly different from the tangent point. Any volunteers to redraw the diagram with a triangle having more widely differing lengths of sides?    D b f i r s   12:01, 28 June 2010 (UTC)

The meaning of inverse altitudes
In the section Formulas mimicking Heron's formula:

Next, denoting the altitudes from sides a, b, and c respectively as $$h_a$$, $$h_b$$, and $$ h_c$$,and denoting the semi-sum of the inverse altitudes as $$H = (h_a^{-1} + h_b^{-1} + h_c^{-1})/2$$ we have


 * $$\mathrm{Area}^{-1} = 4 \sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}.$$

What is the meaning of inverse altitudes? Doraemonpaul (talk) 22:32, 10 August 2010 (UTC)


 * I think, from context, that "inverse" here means "reciprocal". I have fixed the text in the article. Gandalf61 (talk) 08:28, 11 August 2010 (UTC)

Math overload?
The math regarding triangles is great, but there are other aspects too -- use in construction (ie Egyptian pyramids); triangles used for music. What I'm saying is that there are more aspects here which could be added.--Tomwsulcer (talk) 00:41, 27 March 2011 (UTC)


 * This article is about the mathematical figure, and the musical triangle has its own article at Triangle (instrument) with a link from the disambiguation page. Perhaps we could add a new section on the reason for the triangle being fundamental to construction.  Meanwhile, I've moved the image here to be used for this new section, though a picture of a triangular framework would illustrate the point better.  Will you start it or shall I?

 D b f i r s   06:35, 27 March 2011 (UTC)


 * Good idea. I'll start it. I'm somewhat unclear about the math principles why triangles are strong; I know from my handyman work that they're great supports for shelves. I'll start the section. ✅ --Tomwsulcer (talk) 07:21, 27 March 2011 (UTC)


 * Possibly phrasing along the lines of "The triangle is the only polygon in which the shape is uniquely determined by the lengths of the sides, so it is important in construction because a structure that it triangulated cannot change shape (except for bending of the sides of the triangles)". I'm sure this can be improved on.    D b f i r s   08:07, 27 March 2011 (UTC)


 * Yes I see what you're getting at. I tried a wording along these lines; I'm less sure about all the other polygonal shapes out there -- could there be any as strong as a triangle based on shape alone? Wonder what you think of current wording -- can you improve it, wondering.--Tomwsulcer (talk) 16:22, 27 March 2011 (UTC)

Triangle definition
Shouldn't the definition state that a triangle has only or exactly three sides and angles? After all, a square also has three sides and angles. The article's opening sentence reads: "A triangle is ... a polygon with three corners or vertices and three sides...." The definition at thefreedictionary.com states that a triangle is the plane figure formed by connecting three points not in a straight line by straight line segments; a three-sided polygon. Urgos (talk) 23:31, 17 June 2012 (UTC)
 * This sounds reasonable to me, but it would be inconsistent with the other polygon articles, so I'm not sure. I would look to see if this has been discussed anywhere else first. Nat2 (talk) 22:45, 25 July 2012 (UTC)

Exclusive definition of isosceles triangle
Is the exclusive definition of isosceles triangles (i.e. that equilateral triangles are not isosceles) still taught anywhere today? I don't think it is, and if it's not, I think it's being given undue weight in the article. Jackmcbarn (talk) 17:00, 4 January 2014 (UTC)


 * Yes, I think it is still taught, just as some American schools teach the exclusive definition of the trapezium (trapezoid). I agree that inclusive definitions are preferable.    D b f i r s   22:11, 4 January 2014 (UTC)


 * Alas, it is taught, as I know only too well after decades of teaching mathematics in high schools. Would that it weren't. However, the article currently says "Some mathematicians define an isosceles triangle to have exactly two equal sides". Are there any mathematicians who define it that way, as opposed to school teachers? Is there a source for that? The article gives a source for the (as far as I know) more standard definition, where equilateral ∈ isosceles, but none for the other definition, and I personally have no memory of ever seeing the exclusive definition given by any serious mathematical source. JamesBWatson (talk) 22:13, 4 January 2014 (UTC)
 * Unfortunately, that's how Euclid's Elements defines it: "an isosceles triangle that which has two of its sides alone equal". Jackmcbarn (talk) 22:39, 4 January 2014 (UTC)
 * That is interesting, but not relevant here, because how a Wikipedia article should use a term is determined by the normally understood usage in early 21st century English, not 4th/3rd century BC Greek. JamesBWatson (talk) 21:16, 6 January 2014 (UTC)
 * The problem is that Euclidean geometry is still taught using Euclid's terms. Charles Lutwidge Dodgson wrote a book on Euclidean geometry ( I've read it. ) and also wrote:
 * '' "When I use a word," Humpty Dumpty said in a rather a scornful tone, "it means just what I choose it to mean – neither more nor less."
 * '' "The question is," said Alice, "whether you can make words mean so many different things."
 *  "The question is," said Humpty Dumpty, "which is to be master – that's all."
 * My point being that there are two different "normally understood usage[s] in early 21st century English".
 * For a source, try math.com lessons (though another lesson contradicts this usage).   D b f i r s   23:11, 6 January 2014 (UTC)
 * The exclusive definition seems also to be implicit in mathopenref and is explicit in "Isosceles triangles have two sides with the same length, and one side that differs." that Google claims to find on the freemathhelp.com website (though I can't see those words there). I'm sure I can find some British school textbooks that use Euclid's definition, but they tend not to be published on the internet.    D b f i r s   23:30, 6 January 2014 (UTC)
 * Yes indeed. There are certainly British school text books that give the exclusive definition, jsut as there are British school textbooks that state that 1 is a prime number. My question is whether ther are any mathematicians who use the term in that way, and I don't count someone who got a grade E at A level maths, went to a college to train to be a teacher, and after a few years of teaching decided to write a text book as a "mathematician". Is there any evidence of the use of the exclusive definition by serious academic mathematicians? JamesBWatson (talk) 12:26, 7 January 2014 (UTC)
 * I haven't seen the prime number error in British text books, but Euclidean geometry is still a valid branch of mathematics, so certainly Charles Dodgson used the exclusive definition. I don't think modern mathematicians worry about such trivialities.  They don't write papers on such basic geometry.   D b f i r s   14:45, 7 January 2014 (UTC)

Phrasing of Scalene Definition
The sentence "Right triangles are scalene if and only if not isosceles." sounds awkward and might even be incorrect, like it's saying a right triangle is isosceles and not isosceles. I think a better phrasing would be the simpler, "A right triangle is scalene only if it is not isosceles." — Preceding unsigned comment added by Threefour (talk • contribs) 04:13, 6 June 2014 (UTC)
 * The statement as given is true, but has nothing to do with right triangles. Any triangle is scalene if and only if it is not isosceles (one must consider equilateral triangles as isosceles for this statement to be correct - see above section). Your modification is simpler only because it throws out half of the statement being made. See If and only if for clarification of this linguistic construction. Bill Cherowitzo (talk) 04:47, 6 June 2014 (UTC)
 * Well the statement is true if and only if you use the inclusive definition of isosceles. I share Threefour's dislike of the form of the statement.  How can we re-phrase it?    D b f i r s   06:33, 6 June 2014 (UTC)
 * Sorry, early in the morning here, and my brain wasn't in gear!   D b f i r s   06:43, 6 June 2014 (UTC)

Area by line integral
The section on getting the area specifies how to obtain the line integral between two consecutive vertices, but then doesn't specify how to go from that to actually get the area. The page on line integrals that is linked to is rather technical, which makes it difficult to find out how to use this method.

FreeFull (talk) 14:00, 4 August 2015 (UTC)

Semi-protected edit request on 6 December 2015 - misspelling
change "hypontenuse" to "hypotenuse" (find/replace)

Cjnat (talk) 20:01, 6 December 2015 (UTC)
 * Yes check.svg Done RudolfRed (talk) 20:15, 6 December 2015 (UTC)

Semi-protected edit request on 4 January 2016
"The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side."

It's not true. Lengths of any two sides of a triangle must be greater to the length of the third side. It can not be equal. We can't build triangle from lengths 2, 3 and 5. So it should be like:

"The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater to the length of the third side."

194.11.254.132 (talk) 12:25, 4 January 2016 (UTC)


 * Request rejected. The triangle inequality is correctly stated. You can indeed have a triangle with side lengths 2, 3 and 5 - it just happens to be a degenerate triangle with zero area. Gandalf61 (talk) 12:49, 4 January 2016 (UTC)


 * This is already explained in the article. (I share the OP's dislike of degenerate polygons.)    D b f i r s   13:54, 4 January 2016 (UTC)
 * Do you accept self-intersecting polygons, though?
 * If so, then we have an interesting conundrum. Consider a square ABCD. Now we move A and B closer together on the line passing through them until they coincide, forming what looks like an isosceles triangle. Then keep moving them the same way until they swap places. If one accepts self-intersecting polygons but not degenerate polygons with partially or completely coinciding elements, then every polygon along this sequence is legitimate – except for the transition between the self-intersecting and non-self-intersecting cases, which is not. This seems to me slightly distasteful. Double sharp (talk) 15:37, 4 January 2016 (UTC)
 * ... so you are asking "can a triangle be a square if we give two names to one vertex?"
 * I dislike degeneracy, and regard self-intersecting polygons as "not the real thing", but I accept that one can define a polygon more generally than my rather restrictive preference.   D b f i r s   17:34, 4 January 2016 (UTC)

Semi-protected edit request on 22 February 2016
For the graphic https://en.wikipedia.org/wiki/File:Triangle.TrigArea.svg used under the headings "Computing the area of a triangle" and "Using trigonometry", should it not include the label "c" for side AB?

64.185.148.174 (talk) 04:32, 22 February 2016 (UTC)
 * The author presumably designed the image for the trigonometry where c is not mentioned, so perhaps deliberately omitted labelling that side. It isn't possible to edit the image from within the article.  It was created by editor Limaner who hasn't edited since last April.  It would be possible to create a new image if anyone thinks it worthwhile.    D b f i r s   08:27, 22 February 2016 (UTC)

Properties of circles and triangle
I assume all triangles have properties and they should be classified in this article as seven,and circles have the same properties concerning angles without the use of calculator. examples: there are 7 trianlge in which the perimeter are:
 * $$5+5+1=11$$
 * $$5+5+2=12$$
 * $$5+5+3=13$$
 * $$5+5+4=14$$
 * $$5+5+5=15$$
 * $$5+5+6=16$$
 * $$5+5+7=17$$


 * $$1-0.98=0.02$$
 * $$1-0.92=0.8$$
 * $$1-0.82=0.18$$
 * $$1-0.68=0.32$$
 * $$1-0.5=0.50$$
 * $$1-0.28=0.72$$
 * $$1-0.02=0.98$$

so 11+12+13+14+15+16+17=98 and 98/7=14

there are 7 circles in which the sum of the angles equals 98


 * $$\arccos0.98+\arccos0.1+\arccos0.1=180$$
 * $$0.98+0.1+0.1=1.18$$
 * $$\arccos0.92+\arccos0.2+\arccos0.2=180$$
 * $$0.92+0.2+0.2=1.32$$
 * $$\arccos0.82+\arccos0.3+\arccos0.3=180$$
 * $$0.82+0.3+0.3=1.42$$
 * $$\arccos0.68+\arccos0.4+\arccos0.4=180$$
 * $$0.68+0.4+0.4=1.48$$
 * $$\arccos0.5+\arccos0.5+\arccos0.5=180$$
 * $$0.5+0.5+0.5=1.5$$
 * $$\arccos0.72+\arccos0.6+\arccos0.6=180$$
 * $$0.28+0.6+0.6=1.48$$
 * $$\arccos0.02+\arccos0.7+\arccos0.7=180$$
 * $$0.02+0.7^2+0.7^2=1.42$$

seven
 * $$1.18+1.32+1.42+1.48+1.5+1.48+1.42=9.8$$
 * $$2+6+10+14+18+22+26=9.8$$


 * $$0.68+0.4+0.4=1.48 \arccos0.68+\arccos0.4+\arccos0.4=180$$
 * $$0.68+0.16+0.16=1$$
 * $$1-0.68=0.32$$
 * $$\frac{0.32}{2}=0.16, \sqrt0.16=0.4$$
 * $$0.68 ,0.68+0.16=0.84, 0.84+16=1$$
 * $$0.68\times100=68,0.84\times100=84,1\times100=100$$
 * $$\frac{68}{4}=17 \frac{84}{4}=21 \frac{100}{4}=25$$

199.7.157.124 (talk) 12:38, 18 July 2016 (UTC)199.7.157.105 (talk) 20:00, 16 July 2016 (UTC)

Diagram of isosceles right triangle
I have just relabeled this newly added diagram, but I fail to see that it really adds anything to the article. Certainly its location is inappropriate. Any comments or suggestions would be appreciated. --Bill Cherowitzo (talk) 17:13, 6 November 2016 (UTC)

Edit request, 28 March 2017
"the latter equality applying only if none of the angles is 90°"

"latter" means the second of two things. Change to "last". 195.157.65.228 (talk) 17:17, 28 March 2017 (UTC)
 * Done, thanks. Paul August &#9742; 17:45, 28 March 2017 (UTC)

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any three non-collinear points determine a unique plane?
Article says: "In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space)." I believe the part about "unique plane" is incorrect, because any plane has an infinite number of triangles. The triangle from the three points is unique, because it is different (different vertices) from any other triangle. But the plane is not unique, because an infinite number of other triangles determine the same plane. If I misunderstand, someone please correct me. The correct wording, IMO, should be "In Euclidean geometry any three points, when non-collinear, determine a unique triangle." AAABBB222 (talk) 18:04, 14 August 2017 (UTC)


 * The statement in the article is quite correct, three non-collinear points do determine a unique plane. The fact that there are many triangles in a plane does not change this (consider this − two distinct points determine a unique line, but there are an infinite number of pairs of points on this line.) One of the easiest ways to see that this is true is to use analytic geometry (coordinates) and some linear algebra. --Bill Cherowitzo (talk) 18:38, 14 August 2017 (UTC)


 * To me it seems incorrect. In addition to what I already said, the article is about triangles, not planes. So why start the article with a statement about planes that is confusing at best? Furthermore, you can reverse the logic to say "In Euclidean geometry any triangle determines a unique set of three non-collinear points." However, trying to do the same for the plane statement fails: "In Euclidean geometry any plane determines a unique set of three non-collinear points." :( Finally, the word "determine" is a synonym for "define" here, and a triangle does not "define" a plane. The furthest I would go and keep the explanation clear is something like "A triangle lies on one, and only one, plane". AAABBB222 (talk) 21:38, 14 August 2017 (UTC)


 * I'm sorry, but your intuition is leading down the wrong path. A triangle does define a plane and your final statement is just another way to say that a triangle determines a unique plane. There are reasons to bring up planes in regards to triangles. To start with, it should be made clear that triangles are objects that live in planes. The same can not be said for sets of four or more points. Also, when you move away from Euclidean geometry, the statement about triangles living in planes is no longer always true, and this is brought up in the article. I have expanded that paragraph in the article somewhat, but I am not sure that this will satisfy you. --Bill Cherowitzo (talk) 23:18, 14 August 2017 (UTC)


 * I think we would be better served not making comments about my intuition, and sticking with the facts. A square also defines a plane. So does a pentagon, hexagon, etc. So I think it is not needed, and confusing, to introduce a triangle by suggesting it is somehow special because it "determines" a plane. The other polygon articles don't waste space on that, so neither should this one (Please do not go editing them to say they "determine a unique plane"). And there is more than three points to the concept of a plane, so I think it is incorrect to say that a triangle "defines a plane". It simply lies on a plane. And again reversing the logic of the statement does not work, as pointed out in previous comment, so the statement is confusing. A triangle is simply a three-sided polygon, period. Maybe I should have just edited the article. Anyone else? AAABBB222 (talk) 21:32, 23 August 2017 (UTC)


 * A square, pentagon, hexagon, etc. are all defined as plane figures, so yes they do determine a plane–the one they live in. However, in Euclidean 3-space, the four vertices of a quadrilateral do not determine a plane, nor do the correct number of vertices for any of the spacial polygons, except for the triangle. That is why it is important to bring this up. That three non-collinear points define a plane has a very clear mathematical meaning. See Plane (geometry) for the details. Your "reversing the logic" argument makes no logical sense. You are simply interchanging the words triangle and plane as if they are identical objects. A triangle determines a unique plane, but a plane contains many triangles (any one of which determines that plane). If all of geometry was limited to the plane, then your point might have some merit, but when higher dimensions are involved you must make allowances for the fact that not all planar concepts are still valid. --Bill Cherowitzo (talk) 22:14, 23 August 2017 (UTC)
 * I can only agree with Bill. Sometimes mathematical English makes use of constructions that do not sound very normal in non-mathematical English, but to get anywhere you have to know what is being meant. Double sharp (talk) 23:52, 23 August 2017 (UTC)
 * I agree that the article is accurate as it stands, but we will be happy to consider any improvements to avoid misunderstandings of what is stated there. Please discuss them here first.   D b f i r s   08:08, 24 August 2017 (UTC)

I respectfully disagree. A triangle is simply a three-sided polygon, period. That is the necessary and sufficient definition, stripped of filler and verbiage. AAABBB222 (talk) 23:01, 13 February 2018 (UTC)
 * Which is, in fact, given in the first sentence: "A triangle is a polygon with three edges and three vertices". But surely the lede should not consist of one sentence and some more context is desirable. Double sharp (talk) 23:40, 13 February 2018 (UTC)

TRIANGLE : 5.3 Solution of triangles
About solution of triangles, 5.3 the next sentence could be more accurate if the last section is added:

"Solution of triangles" is the main trigonometric problem: to find missing characteristics of a triangle (three angles, the lengths of the three sides etc.) when at least three of these characteristics are given , including at least one non-angular value.

Could somebody make this appropriate correction ?

--Duckkcud (talk) 19:51, 7 April 2018 (UTC)Duckkcud 2018-04-07


 * While appropriate for planar triangles, it is not appropriate for spherical triangles since AAA is a congruency condition there.--Bill Cherowitzo (talk) 03:17, 8 April 2018 (UTC)

In the introduction, it is written: "This article is about triangles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted."

For solution of non-Euclidian triangle, you will need more than AAA; you need R ( a non angular value ) to find the length of the sides... In the end exemple of non-planar geometry, the position of the corners give the ratio of surface, but still not the length of the sides... Duckkcud 2018-04-10 — Preceding unsigned comment added by Duckkcud (talk • contribs) 17:05, 10 April 2018 (UTC)


 * The section specifically includes triangles on a sphere, so any statement made should also be applicable to that case. Also, in the spherical case, the lengths of the sides of the triangle are not measured in linear units (to do so you would need to know $R$ the radius of the sphere, as you correctly point out) but rather in angular units referring to the central angles determined by the sides and this does not require knowledge of $R$. This, probably, is due to the astronomical applications, where $R$ can not be accurately determined. See the main article Solving Triangles for more information on this. The example in the non-planar triangle section of this article is not germane to this question.

As this article on "Triangles" deals primarely on plane one, it should be normal to precise its properties without confusion. That is exactly precised in the article on "solution of triangle" in wiki: '...at least one of the side lengths must be specified...' https://en.wikipedia.org/wiki/Solution_of_triangles#Solving_plane_triangles The surface, the perimeter can too serve to solution. But only angles, are not enough. http://www.rlefebvre.ca/triangulateur/triangulator.htm In the paragraph that I propose to correct, there is no reference to spherical triangle. So, it is an error to not mention what I propose. DUckkcud 2018 04 11 — Preceding unsigned comment added by Duckkcud (talk • contribs) 13:51, 11 April 2018 (UTC)
 * The section, in its entirety is:

"Solution of triangles" is the main trigonometric problem: to find missing characteristics of a triangle (three angles, the lengths of the three sides etc.) when at least three of these characteristics are given. The triangle can be located on a plane or on a sphere. This problem often occurs in various trigonometric applications, such as geodesy, astronomy, construction, navigation etc. What part of "The triangle can be located on a plane or on a sphere." do you not understand? Your statement is valid for planar triangles, and I have said so above, so I am not sure what you intend by bringing up sources which only refer to planar triangles.--Bill Cherowitzo (talk) 18:25, 11 April 2018 (UTC)

I should have written "in the sentence", not "in the paragraph". And calling an error was not appropriate. Excuse me. My intent is to take the occasion to say exactly how many and what inputs are necessary to solution a planar triangle without confusion with spherical one. "At least three" is a bit vague while the number 3 is the beauty of triangle. I still suggest: "when three of these characteristics are given, including at least one non-angular value on a plane." --Duckkcud 2018 04 11 — Preceding unsigned comment added by 208.114.128.61 (talk) 21:24, 11 April 2018 (UTC)

Semi-protected edit request on 22 August 2019
in the "See also" section add "STL (file format)". wikipedia.org/wiki/STL_(file_format) Austinthemighty (talk) 06:22, 22 August 2019 (UTC)
 * Red information icon with gradient background.svg Not done: triangulation is already mentioned as a more general case so specific file format does not seem necessary. Melmann 10:03, 22 August 2019 (UTC)

Adding the area of a cevian triangle
Hi guys,

Read the "Be Bold" statement so will add a short entry under "Figures inscribed in a triangle" for cevian triangles then pursue a separate entry for this subject.

Dominic — Preceding unsigned comment added by 108.167.57.225 (talk) 12:01, 3 November 2019 (UTC)


 * Just remember to follow what published reliable sources have said, we don't publish original research or original mathematical proofs and we can't cite forums. – Thjarkur (talk) 12:11, 3 November 2019 (UTC)

For make Fe-thriangle what will needs ?
Electric welding. And three electrodes! .. ))) 176.59.194.139 (talk) 09:45, 24 May 2020 (UTC)

Correction needed?
I think that the figure labeled "Euler diagram of types of triangles" should be changed because there is not a triangle isosceles and obtuse at the same time.

Jon Peli Oleaga (talk) 20:01, 22 December 2020 (UTC)


 * Perhaps you are thinking of something else. A triangle with angles 120°-30°-30° is both obtuse and isosceles.--Bill Cherowitzo (talk) 20:07, 22 December 2020 (UTC)

You're right; I have tried to correct my mistake, but you caught it before I could. Jon Peli Oleaga (talk) 20:14, 22 December 2020 (UTC)

Semi-protected edit request on 26 September 2021
I just Want to add some very important things. Thank you, 2405:204:109F:48F:E50E:7CE4:71BC:495A (talk) 06:22, 26 September 2021 (UTC)
 * Red information icon with gradient background.svg Not done: this is not the right page to request additional user rights. You may reopen this request with the specific changes to be made and someone will add them for you, or if you have an account, you can wait until you are autoconfirmed and edit the page yourself. — Sirdog (talk) 06:44, 26 September 2021 (UTC)

'''A triangle is a closed shape with 3 angles, 3 sides, and 3 vertices. A triangle with three vertices says P, Q, and R is represented as △PQR. It is also termed a three-sided polygon or trigon. In this mini-lesson, we will explore everything about triangles, which are commonly seen around us. If you observe the shape of signboards and your favorite sandwiches it forms the shape of a triangle. ''' — Preceding unsigned comment added by Huligevva H (talk • contribs) 08:34, 9 February 2022 (UTC)

Semi-protected edit request on 23 June 2021
Add an external link at the end :

Triangulator Calculator for any triangle from a minimum of data; with a drawing of the triangle. Duckkcud (talk) 16:27, 23 June 2021 (UTC)
 * Red information icon with gradient background.svg Not done for now: please establish a consensus for this alteration before using the template. That link seems really unnecessary and (while not as big of a deal) is kind of a mess and somewhat hard to use.  Bsoyka  ( talk  &middot;  contribs ) 20:13, 23 June 2021 (UTC)

Integral From Wikipedia, the free encyclopedia Jump to navigationJump to search This article is about the concept of definite integrals in calculus. For the indefinite integral, see antiderivative. For the set of numbers, see integer. For other uses, see Integral (disambiguation). "Area under the curve" redirects here. For the pharmacology integral, see Area under the curve (pharmacokinetics). Definite integral example A definite integral of a function can be represented as the signed area of the region bounded by its graph. Part of a series of articles about Calculus Fundamental theorem Leibniz integral rule Limits of functionsContinuity Mean value theoremRolle's theorem Differential Integral Lists of integralsIntegral transform Definitions AntiderivativeIntegral (improper)Riemann integralLebesgue integrationContour integrationIntegral of inverse functions Integration by PartsDiscsCylindrical shellsSubstitution (trigonometric, Weierstrass, Euler)Euler's formulaPartial fractionsChanging orderReduction formulaeDifferentiating under the integral signRisch algorithm Series Vector Multivariable Specialized Miscellaneous vte In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,[a] and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.

Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs.

Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space. — Preceding unsigned comment added by Huligevva H (talk • contribs) 08:45, 9 February 2022 (UTC)

Finding an unknown distance
Distance 129.0.212.221 (talk) 14:26, 12 November 2022 (UTC)

Confusion what is a triangle
The lede seems to exclude degenerate triangles as triangles, but the article body seems to include degenerate triangles in the class of triangles. Maybe trhe mention of colinearity in the lede should be dropped, as it gives trhe false impression that colinear points do not result in a triangle. 37.49.68.13 (talk) 14:13, 2 March 2022 (UTC)


 * There are several possible concepts of "triangle" which should perhaps be mentioned. For example, a triangle can be taken as a configuration of three points, a configuration of three coplanar lines, a set of three side lengths satisfying the triangle inequality, a (cyclically) ordered triple of three vectors summing to zero, a triple of (signed or unsigned) external angles composing to a full turn, the intersection of three half-planes, ...
 * The definition chosen will determine whether various kinds of degenerate triangles are well defined or not (three colinear points form one kind of degenerate triangle; a half-infinite strip with two right angles and a third vertex at infinity is another kind of degenerate triangle). –jacobolus (t) 17:39, 12 November 2022 (UTC)

Semi-protected edit request on 20 March 2023
Fix the Wiktionary link in the Notes section; the capital "i" breaks it, and should be replaced with https://en.wiktionary.org/wiki/isosceles_triangle. Thanks! 138.229.250.70 (talk) 06:03, 20 March 2023 (UTC)
 * ✅ Good catch, thanks. Actualcpscm (talk) 08:33, 20 March 2023 (UTC)

Semi-protected edit request on 29 March 2023
change Non-planar triangles to Non-planar triangles 141.226.120.166 (talk) 09:13, 29 March 2023 (UTC)
 * by reverting a recent heading change. D.Lazard (talk) 11:02, 29 March 2023 (UTC)