Talk:Triangle inequality

Proof of converse

A recent thread in the Math help desk called to attention some issues with the proof of the converse given here. I realize the proof was asked for in a previous comment, but the proof given is unsourced and seems somewhat ORish. Here are some specific issues.

• In the second paragraph the proof seems to be assuming the conclusion. It fact it's not but it finds conditions necessary for a point to be the vertex of the required triangle, then finds a point that meets this condition. You would then have to work backwards to show that the point is the vertex of the required triangle. The proof could be rephrased to put it the correct order.
• The proof assumes the existence and properties of complex numbers which seems unnecessary since it's a theorem in Euclidean geometry. In fact it's given in The Elements as Book 1 Prop. 22.
• The proof assumes Cartesian coordinates which in turn assumes the parallel postulate. Euclid does not use the parallel postulate so presumably the theorem would still be true in hyperbolic geometry.
• Most of the proof involves computing the coordinates of the vertex which is the intersection of two circles. But for a geometric construction you really only need to specify the circles and label the point where they meet, assuming of course that conditions are met which guarantees such a point.

As I see it there are several alternatives:

• Drop the proof entirely and put it a reference for it; since it's in The Elements it shouldn't be hard to find one.
• Find a source for the existing proof and rephrase to put it a more logical form.
• Replace the proof with Euclid's. (Euclid's proof has some issues as well, but the necessary interpolations can be found in Heath's commentary.)
• Source and rephrase the existing proof and give Euclid's version as an alternative.

Here is my version of Euclid's proof with modifications given (mostly) as in Heath:

Proposition: Given lengths a, b, and c with a+b>c, a+c>b, b+c>a, construct a triangle with sides a, b and c.

First, relabel if necessary so that c is at least as great as a and b. (This step is my own but it saves having to go into cases as in Heath's version.) Then a, b ≤ c < a+b. On line DE, starting at D, mark off DF=a, FG=b and GH=c. Draw circle S with center F and radius a, and circle T with center G and radius c. From G, mark of GM=c in the direction of D. At this point the diagram looks something like:

  D  M F      G        H  E
--*--*-*------*--------*---
  |  a |   b  |    c   |

The point M lies on T and b ≤ c < a+b so GF ≤ GM < GD and M is on the segment DF. Then a+b>c=GM=GF+FM=b+FM so FM<a and M lies inside circle S. The point H is also on T and FH=b+c≥b+a>a so H lies outside the circle S. Since S has points both inside and outside of T, it must have a point on T, say K. (Here Heath invokes what he calls the "Principle of Continuity", not to be confused with Leibniz' Law of Continuity.) But FK = a, FG=b and GK=c so FGK is the required triangle.

Euclid's original proof is much shorter since it simply assumes that the circles S and T will intersect; in this way the proof does not seem to use the necessary hypotheses.

Anyway, what options to people prefer? Is there a source for the proof given or a similar modern style proof which can be given? --RDBury (talk) 06:11, 18 February 2017 (UTC)

• I think you should resequence the current proof more to your satisfaction. I thought I fixed it yesterday with a slight rewording, but if you don't think it's okay now please go ahead and resequence it.
• I don't see where the current proof assumes the existence and properties of complex numbers. It just says that h exists because we are not taking the square root of a negative number.
• The proof says it uses the Cartesian plane but doesn't really. I'll replace it with Euclidean plane, and get rid of the superfluous coordinates.
• The proof doesn't involve computing any coordinates, just the length h. (His notation (d, h) need not be taken as coordinate notation, just as a pair of numbers being computed.)
• I like the current proof (and particularly how it factors out the radicand), and think it should be retained. I'll look for a source, but my access to references is very limited so I'm not too optimistic about finding one. If you like and can put in a good diagram, you could put in your proof as well.

Loraof (talk) 19:05, 18 February 2017 (UTC)

I did find a source for a similar proof, see . The factorization of the radicand is interesting and should be mentioned, but the same factorization appears in in Heron's formula. Maybe it makes more sense to mention that it's the triangle inequality which guarantees that the radicand is positive. My issue with the use of coordinates is that they require the parallel postulate, but even it you remove them you would still be using the Pythagorean theorem which also uses the postulate, and it seems significant that the converse would hold in the non-Euclidean plane. Also, if you remove coordinates you have to contend with the possibility that d is negative, which means you'd need to use signed lengths or split into cases.

In any case, I gather that you'd like to see both proofs in the article, which I don't have a problem with if the issues are fixed. The proof above should probably be trimmed some before it goes in the article; I'll think about how best to do that. --RDBury (talk) 07:12, 19 February 2017 (UTC)

hypervolume and hyperarea

@JayBeeEll: thank you for your recent edits. One quibble, though. I would like to refer to the measure of an (n − 1)-dimensional facet as a "hyperarea" rather than a "hypervolume". Specifically, I use "hypervolume" as the general term for the measure of a "face" of a structure (e.g., a simplex), where a face can be of any number of dimensions relative to the number of dimensions of the structure. However, a "facet" is defined to be a face of one less dimension than the full structure. When one is referring specifically to this number of dimensions, it is nice to have this more-specific word to use. Likewise, with "facet" comes "hyperarea". When referring specifically to this number of dimensions, it is nice to have this more-specific word to use.

Obviously, the more general words could be used instead of "facet" or "hyperarea". However, math is about precision and I'd like to use the more precise "hyperarea" in this case. —Quantling (talk | contribs) 13:33, 24 October 2021 (UTC)

I understand your motivation, but as far as I can tell this is a word you just made up for convenience, not one that is represented in reliable sources. In particular, I do not believe there is a meaningful concept to be attached to the phrase "hyperarea of [a thing]". In contrast, length, area, and volume have fixed meanings, and they are all particular cases of hypervolume (in appropriate dimensions). As evidence in favor of the theory that this is not a real thing, the word "hyperarea" is used in only two other articles on Wikipedia: Law of sines (it was added in this edit, with no reference supporting its use) and Polyakov action (where it does not appear to be used with the same meaning). If it were a term with an accepted meaning used by some relevant community, I would expect it to be much better-represented in articles on polyhedra etc. --JBL (talk) 16:57, 24 October 2021 (UTC)

I don't see much use of "hyperarea" outside of Wikipedia either. (Though, not zero. If I've made it up, so have others!) We should probably do this your way; including correcting the Law of sines Wikipedia page. —Quantling (talk | contribs) 23:14, 24 October 2021 (UTC)

Before we put this issue completely to bed, what do you think of "hyper-surface area" (instead of "hypervolume") where we are replacing "hyperarea"? The wording "hyper-surface area" gets a few relevant search engine hits. —Quantling (talk | contribs) 16:10, 25 October 2021 (UTC)

Distances, points, vectors

The article is a little sloppy when discussing distances and vectors. Specifically, the triangle inequality for distances is either z ≤ x + y for distances x, y, and z or, if the distances are between specified points, something like d(a, c) ≤ d(a, b) + d(b, c). On the other hand, if u is the vector from point a to point b and v is the vector from point b to point c then the triangle inequality for vectors is |u + v| ≤ |u| + |v|. Let's have all three of these formulas, properly labeled. —Quantling (talk | contribs) 20:30, 17 June 2024 (UTC)

The lead of the article seems very clear; what part are you unhappy about? --JBL (talk) 00:47, 18 June 2024 (UTC)

I will make an edit to the article. —Quantling (talk | contribs) 12:34, 18 June 2024 (UTC)

x, y, z vs. a, b, c

The lede uses x, y, and z for distances, but the remainder of the article uses a, b, and c; shouldn't we be consistent? For vectors the article uses x and y consistently, but that kind of, sort of collides with the notation for lengths; should we make those more distinct? Likewise, when we get to metric spaces, now x, y, and z are points; should we have a third set of letters for these? In summary, we have: $$z\leq x+y$$ $$\|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|$$ $$d(x,z)\leq d(x,y)+d(y,z)$$ but IMHO the following would be better:

${\textstyle c\leq a+b}$ for lengths a, b, and c;

${\textstyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|}$ for vectors u and v; and

${\textstyle d(P,R)\leq d(P,Q)+d(Q,R)}$ for points P, Q, and R.

I'd make changes myself, but at least one of the figures depends upon the current mishmash of variable names, and I don't have the skills to edit that. —Quantling (talk | contribs) 12:53, 18 June 2024 (UTC)

I like a, b, and c for lengths because of their use in the common presentation of the Pythagorean theorem a² + b² = c². —Quantling (talk | contribs) 12:56, 18 June 2024 (UTC)

I edited the article, except using A, B, and C for points. —Quantling (talk | contribs) 16:19, 19 June 2024 (UTC)

Are you going to change the diagrams? –jacobolus (t) 17:03, 19 June 2024 (UTC)

No, unfortunately. In that sense, I've changed the article from one inconsistent state to a different inconsistent state :frown:. I think it is now slightly less inconsistent and is less confusing (no reuse of a symbol to mean multiple things) — but updating the diagrams needs to happen too! In the meantime, I left the captions consistent with their figures. —Quantling (talk | contribs) 17:33, 19 June 2024 (UTC)

​Geometric progression of polygon side lengths (reverting revision 1377854198)

Hi,

@Quantling

I’d like to discuss the recent revert of the addition concerning polygons whose side lengths form a geometric sequence (revision 1377854198).

The edit summary stated:

"r=1.9 with sides (1, 1.9, 3.61) fails the triangle inequality despite r ∈ (1/2, 2)"

I believe this revert was based on a misunderstanding of the theorem. The statement was not that every n-gon (e.g. a triangle, n = 3) exists for any r \in (0.5, 2), but rather that for any ratio r \in (0.5, 2), there exists an n-gon provided the number of sides n is sufficiently large.

For r=1.9 the lower bound is 4.59, so indeed no triangle can be formed.

That being said, the text contained a typo (double ln, where there should be a single ln).

Could we reinstate the text with the correction?

I'd be happy to submit it if we are in agreement.

Thanks ~2026-52850-53 (talk) 18:21, 1 October 2026 (UTC)

The first sentence of the text that I deleted was:

In the general case of an $n$-gon, a necessary and sufficient condition for the existence of such an $n$-gon whose side lengths form a geometric sequence with common ratio $r$ is that $r$ belongs to the interval $(0.5,2)$.

It could be interpreted to be saying that ∀n ≥ 3, ∀r ∈ (0.5, 2), ∃ a polygon with n sides and geometric ratio r. We'd have to rewrite this to make it clear that we mean ∀r ∈ (0.5, 2), ∃n ≥ 3 s.t. ∃ a polygon with n sides and geometric ratio r. or something similar.

However, we must also overcome WP:UNDUE and WP:OR. Do you have a citation to a noteworthy textbook (or similar reliable source) that indicates that this is a useful derivation? If not, I don't see that it is worthy of all the space that we are giving it. —Quantling (talk | contribs) 19:24, 1 October 2026 (UTC)

Oxman, Victor. (2024). "On a polygon whose side lengths form a geometric sequence”, International Journal of Mathematical Education in Science and Technology, 57(3), 591–597. https://doi.org/10.1080/0020739X.2024.2404416

This article (referenced from the text) is published in a reputable journal, and is a reliable source.

Re the text, there are two inextricably linked sentences, and the necessary and sufficient condition cannot be interpreted without taking the second sentence into account. ~2026-52850-53 (talk) 10:45, 2 October 2026 (UTC)

Of all the things we might say about the triangle inequality, this seems like a fairly unimportant one. I'd leave it out. We don't need to cover obscure new papers when we're ignoring hundreds or thousands of more relevant and more widely cited papers from decades ago. –jacobolus (t) 12:45, 2 October 2026 (UTC)

I'd just as soon remove the whole section § Example of the generalized polygon inequality for a quadrilateral. –jacobolus (t) 12:47, 2 October 2026 (UTC)

Keeping the lead accessible

The lead has what I consider to be a very inaccessible description of the triangle inequality for vectors.

In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms):

$$\|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|,$$

where the length of the third side has been replaced by the length of the vector sum u + v.

I rewrote this for accessibility:

The triangle inequality implies a related statement for Euclidean vectors, also called the triangle inequality; if u and v are vectors (directed magnitudes, usually drawn as arrows), then u, v, and their sum u + v can be pictured as a triangle with arrows for sides, with each vector's norm as the length of the corresponding side. Therefore

$$\|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|.$$

In particular, Euclidean vector is a much more useful wikilink than Euclidean geometry which doesn't even mention vectors, the previous version doesn't say what a vector is or why triangles are involved, and it's entirely unclear why you would "replace" the side of a triangle with a length of a vector sum.

User:Quantling reverted this on the grounds that "A triangle that works has origin to the tip of u to the tip of u+v back to the origin, but this text doesn't explain that, so reverting this part for now". I don't understand what "A triangle that works" is supposed to mean. I think the replacement text seems significantly more accessible than the original, and elaborate details of how vector addition works don't belong in the lead; we're just trying to get the basic idea across in an accessible fashion, and further details can be explained in the article body.

I also removed a sentence about real numbers, which Quantling restored:

When u and v are real numbers, they can be viewed as vectors in $\mathbb {R} ^{1}$, and the triangle inequality expresses a relationship between absolute values.

I think this sentence sucks. Many readers are going to have no idea what it means to "view a real number as a vector", won't know what ⁠$\textstyle \mathbb {R} ^{1}$⁠ means, and will not be able to understand what "relationship between absolute values" is implied. For anyone who does understand this, it's a trivially obvious consequence, and anyone who doesn't already understand it won't get anything from the statement. It's also in my opinion out of scope for the lead (but it can be discussed in the body). I think it should be removed.

If someone wants to take a different stab at rewriting the paragraph about vectors, please go ahead. –jacobolus (t) 19:54, 1 October 2026 (UTC)

Yes, I think a rewrite is called for, thank you for starting that. My objection is to then u, v, and their sum u + v can be pictured as a triangle. I think that is confusing, but I couldn't figure out how to fix it off the bat. Ideas? —Quantling (talk | contribs) 19:59, 1 October 2026 (UTC)

I don't think we can or should make a detailed description of what vector addition means in the context of the lead here. We can unpack the idea in the article body. –jacobolus (t) 20:01, 1 October 2026 (UTC)

I restored your new language. I'm still hoping to find language that avoids the word "pictured" or uses that word but also gives a hint at how to picture it. The current language could easily have the novice reader thinking that they should be able to picture this, but failing.

Perhaps if we describe vectors as displacements and indicate that u is an initial displacement via travel in a straight line, v is a subsequent displacement in a straight line, and u + v represents the straight-line displacement that would get to the same place if the displacement were done all at once instead of in two steps. And that is something that I can picture as three sides of a triangle. Yes, that's way too long for the lede — which is why I haven't made the edit to the article — but if you see a way to help the reader see why u, v, and u + v can be interpreted/pictured as the sides of a triangle ... that's what I am aiming for. —Quantling (talk | contribs) 20:23, 1 October 2026 (UTC)

For example, if a novice thinks of u, v, and u + v as points in a plane that are vertices of a triangle they are going down the wrong path. I hope to not give them enough rope to do that. —Quantling (talk | contribs) 20:29, 1 October 2026 (UTC)

Instead of then u, v, and their sum u + v can be pictured as a triangle with arrows for sides how about something closer to then u, v, and their sum u + v can be pictured as sides that are assembled into a triangle. That makes it very hard to think that the vectors might be playing the role of the vertices of a triangle. I'll go boldly edit that. Please undo or, better yet, improve it if you want. —Quantling (talk | contribs) 20:42, 1 October 2026 (UTC)

I'm not too picky. I just think we should give some indication that vectors and their sum are arrows forming the sides of a triangle. I think we should have a dedicated section about this before we get to § Normed vector space which is also pretty inaccessible. –jacobolus (t) 20:50, 1 October 2026 (UTC)

Your version seems fine. –jacobolus (t) 03:25, 2 October 2026 (UTC)

Thinking about this more, I wonder if we should take the part about degenerate triangles entirely out of the lead section or cut it down to a half sentence (moving more discussion to the geometry section), and try to refocus to put more emphasis on the use of the triangle inequality in analysis and other fields.

Overall I don't think this article gives a very accurate impression of where and how the triangle inequality is used in mathematics. I've been looking for sources but I can't find any amazing ones. I don't feel like enough of an expert to do a great job writing about this just off the top of my head (I took undergraduate analysis classes 20 years ago, and haven't done all that much with these types of inequalities since). –jacobolus (t) 19:30, 2 October 2026 (UTC)

To me, the main point of the triangle inequality is its use in theorems such as the one that proves the equivalence of convergent sequences and Cauchy sequences in any metric space (such as ⁠$\mathbb {R}$⁠ or ⁠$\mathbb {C}$⁠). However, that's not terribly accessible to many readers. The stuff about arithmetic or geometric sequences of triangle side lengths is a side show that probably isn't worthy of inclusion in the article, in that hardly any actual mathematicians care, but it is more accessible. How to balance all that .... —Quantling (talk | contribs) 19:59, 2 October 2026 (UTC)

I boldly shortened the "degenerate triangle" part of the lede. —Quantling (talk | contribs) 20:03, 2 October 2026 (UTC)

I was thinking we could shorten this further, to just a phrase at the end of the initial couple sentences that equality implies a degenerate triangle with zero area, and then move more explicit discussion to the § Euclidean geometry section. I think we should remove § Right triangle, which seems like a distracting tangent; we can mention the special case of right triangles in one or two sentences. –jacobolus (t) 20:09, 2 October 2026 (UTC)

Please go bold with those ideas. —Quantling (talk | contribs) 20:13, 2 October 2026 (UTC)

Big chunks of this article have excessive geometrical derivations and detail that feel somewhat irrelevant – I think the authors were perhaps interested in high-school contest problems?

I think we can refocus the geometry section. The part about Euclid's proof is okay (though we should just quote the top-level statement "In any triangle the sum of any two sides is greater than the remaining one." instead of the labeled restatement). Then we can briefly discuss special cases (like degenerate triangles, right triangles, ..>?), then other constraints on valid triangle data. After that I think we can jump to discussing how the triangle inequality can be written in terms of Euclidean vectors, including the generalized triangle inequality with any number of elements (the name "generalized polygon inequality" doesn't seem to be in common use and should be changed), and we can mention how this is related to the shortest distance between two points being a straight line. –jacobolus (t) 20:33, 2 October 2026 (UTC)

I'm not sure whether § Metric space and § Normed vector space should be separate top-level sections. It might be better to combine these. I'm not sure what the right high-level heading titles should be. Maybe it should be something about analysis. The difference in frame for what the "triangle inequality" means in the two are just generalizations the versions of a Euclidean triangle inequality in terms of distances (side lengths) vs. vector magnitude, respectively. –jacobolus (t) 20:46, 2 October 2026 (UTC)