Talk:Heronian triangle

Heronian triangles as integer-sided rather than rational-sided
Although Pythagorean triples could perfectly well be defined as rational triples, the convention has been to define them as integer triples, which has the advantage of being more elementary and therefore reaching a wider audience. Is there some reason I'm overlooking for not following this convention in the case of Heronian triangles, as done e.g. at Dirac Delta? Every Heronian triangle, or for that matter finite set of such, has integer sides and area for some choice of scale. The least situation I can imagine where one would want the more general notion is when dealing with infinite sets of Heronian triangles, which may not have a common denominator. Has this case ever arisen, and even if so, sufficiently often as to warrant depriving the integer-savvy but rational-challenged of an appealing concept? "Things should be made as simple as possible, but no simpler." --Vaughan Pratt (talk) 17:43, 19 November 2008 (UTC)


 * I agree--it is conventional to refer to Heronian triangles as integer-sided rather than rational-sided, since they amount to the same thing but integer-sided is simpler. I think the article should be rephrased to reflect this.  Duoduoduo (talk) 19:04, 24 May 2010 (UTC)

Merger of Heronian triangle and Integer triangle
It seems to me that there is no real justification in having both Heronian triangle and Integer triangle as separate articles. the two concepts are essentially the same, and it would be more helpful to have the information on both in the same place. The other could be either replaced by a redirect, or reduced to a stub, simply defining the concept and linking to the one substantial article. I am much inclined to make this change, but will wait a few days to see if anyone has any objections or other useful suggestions. To avoid duplication or unhelpful splitting of discussion I suggest that any comments be posted to Talk:Integer triangle rather than here. JamesBWatson (talk) 19:49, 22 May 2010 (UTC)


 * See my comments at Talk:Integer triangle. (Why are there two parallel discussion threads about this proposal ?) Gandalf61 (talk) 09:24, 24 May 2010 (UTC)

Exact formula for Heronian triangles
There are a few small misleading statements in Section 3 (Exact formula for Heronian triangles) that can be easily corrected.

1. The claim that "All Heronian triangles can be generated as MULTIPLES of ..." is not quite correct. There are two problems here. The author seems to have changed his point of view and has started talking about "integer triangles with integer areas." In this context the word "multiple" suggests an integer multiple. "Multiple" should be changed to "rational multiple" to make matters absolutely clear. The formula in the Carmichael reference [2] deals explicitly with integral Heronian triangles; his words are

"Every rational integral triangle has its sides proportional to numbers of the form ..., where m, n, k are positive integers and mn > k^2."

2. The absolute values around one factor of c, namely mn-k^2, are superfluous -- mn > k^2 is one of the three constraints.

3. The constraints make no sense to me. It looks as if the author is trying to produce a list of integer triangles that is both complete and without repeated entries. Why? Note that his reference [2] just has m,n,k positive integers (corresponding to the Wiki author's third constraint m \ge n \ge 1) and mn > k^2 (which is the first part of the Wiki author's second constraint). The author's first constraint involving the gcd is harmless -- it just eliminates some duplication. By contrast, the second part of the second constraint, namely k^2 \ge (m^2)n/(2m+n), is unexplained, and my guess is that it is wrong. I see no way to get the example 5, 29, 30 mentioned earlier (which comes from m=36, n=4, k=3, and yields a triangle bigger than the target triangle by a factor of 180). If the constraint is useful and can be explained, a reference for it should be provided. Otherwise one should stick with Carmichael.

By the way, the Carmichael reference is available from ProjectEuclid (where the relevant pages can be read without downloading the entire book) as well as from ProjectGutenberg where the book can be downloaded. Why not provide the link in the bibliography?

Note that these changes would also have to be made on Wiki's "Integer triangle" page, and the definition of "Heronian triangle" there (where these are defined to be integer-sided) should be made consistent with the Heronian triangle page.

FURTHER SUGGESTION: It would be nice if somebody went through both articles making certain that the terminology is consistent. The section on Pythagorean triples is a bit fuzzy -- if the author wants his "Heronian triangles" to have rational sides, then he must explicitly say "integer Heronian triangles" when dealing with integer triangles. (By the way, Pythagorean triangles MUST have integer areas because at least one leg must be even, which is not clear as the entry now stands.) -- Chris Fisher (19 June 2012) — Preceding unsigned comment added by 24.89.64.118 (talk) 04:27, 20 June 2012 (UTC)


 * I have started to update the article to make it consistent. Most sources define a Heronian triangle as an integer triangle with integer area (Weisstein is in the minority in generalising to rational sides and area), so I have used this as the main definition. Gandalf61 (talk) 11:58, 20 June 2012 (UTC)

Rational cosines
I see that the interior angles of a Heronian triangle necessarily have rational sines, e.g., because the area can be written as $(1/2)ab sin C$. Is there a similar, short sweet explanation for why the cosines are rational? The article states that they are rational but doesn't give a proof. — Q uantling (talk &#124; contribs) 23:13, 29 April 2020 (UTC)
 * Ah, yes... Law of Cosines — Q uantling (talk &#124; contribs) 23:16, 29 April 2020 (UTC)

A triangle's side is a line segment. A side's length is a number
apparrently we disagree about whether $(3, 4, 5)$ are "sides" or "side lengths"; perhaps others will chime in here. A result of my edit was to make the article more uniformly use "side length" instead of "side", which the undo restores to a more mixed usage. So, I guess we have at least three options we could weigh in on here; "side", "side length", or some of each. Thank you — Q uantling (talk &#124; contribs) 21:00, 14 July 2022 (UTC)
 * I think that, for instance, "triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle whose sides and area are integers" is perfectly length-unambiguous, and that "correcting" it to "triangles whose side lengths and area are all rational numbers since one can rescale the side lengths by a common multiple to obtain a triangle whose side lengths and area are integers" is unnecessary length-pedantry. Assume that the readers have a minimum of length-intelligence. We don't have to insert length every other length word to be length sure that the readers know they are lengths. It just makes the text length longer, more complicated, and therefore a little harder to read, for no good purpose. Similarly, when we are talking about angles, changing every instance of "angle" to "the measure of the angle" is even more unhelpful, by introducing technical terms "measure" which are probably less familiar, to the target audience (which should not be assumed to be trained mathematicians), than the simple "angle". —David Eppstein (talk) 21:31, 14 July 2022 (UTC)
 * I agree with you that "measure of the angle" is wrong for the article and was a mistake for me to add. I like "side length", which was already in use in much of the article.  (We even have "sidelength" (without the space) in four places, though maybe that should be changed.)  Hopefully, we will get additional opinions here.  — Q uantling (talk &#124; contribs) 15:25, 15 July 2022 (UTC)
 * I tend to find "side length" unnecessarily pedantic, though I can imagine situations where I might employ it for sentence flow or variety of phrasing. XOR&#39;easter (talk) 18:01, 16 July 2022 (UTC)
 * I don’t really have a problem with (where unambiguous) conflating sides with side lengths, but the conflation of angles with angle measures is a serious problem in geometry, especially in mathematics education and material aimed at nonspecialists. There are many possible ways to describe and parametrize angles, orientations, and rotations: a pair of lines or orientations, a pair of vectors, a complex number (sometimes of unit magnitude), a rotation matrix, a half-tangent (stereographic projection of a unit complex number), an unsigned angle measure, a signed angle measure, a signed or unsigned sine, a signed or unsigned chord, etc. In many cases the conflation of angles per se with angle measures leads people to overlook or reason improperly about relationships between angles. It’s a set of ideological blinkers. This is especially problematic when modern readers try to interpret works from the past that were operating under different conventions. Wikipedia could be a lot better on this point, especially in fundamental articles such as angle. For the purpose of this article in particular, a half-tangent or unit complex number is a much more appropriate representation than an angle measure, since angle measures muck up all of the rational relationships with unnecessary transcendental functions. –jacobolus (t) 20:03, 4 December 2022 (UTC)

D.Lazard, can you explain the purpose of your footnote?
The footnote added seems awkward and unnecessary to me:


 * The positiveness of the factors of the right-hand side is a necessary and sufficient condition for $a,b,c$ being the side lengths of a triangle. So, a solution that does not satisfies this condition does not defines a Heronian triangle.

It is known to all readers here that the conventional definitions of length and area are always positive quantities, and therefore only the positive number solutions of Heron's formula are considered to represent valid triangles. Just stating in the text that we care about positive integer solutions seems entirely sufficient.

If this needs to be pointed out explicitly, I thought we might as well additionally note that the signs of a, b, c, A are superfluous insofar as this equation is a polynomial in the squares $$a^2,$$ etc., so for any solution of this Diophantine equation arbitrary sign flips are also solutions. This is at least a nontrivial and kind of interesting insight. So I changed D.Lazard's footnote to instead say:


 * Expanding the Diphantine equation $16\,A^2= 2a^2b^2 + 2a^2c^2 + 2b^2c^2 - 4a^4 - 4b^4 - 4c^4$ reveals it to be a polynomial equation in $a^2,$ $b^2,$ $c^2,$ and $A^2.$ Because $(-x)^2 = x^2$ for any $x,$ if $(a,\,b,\,c,\,A)$ is one solution of this equation, so are all sixteen of the 4-tuples $(\pm a,\,\pm b,\,\pm c,\,\pm A).$ However, conventionally length and area are defined to be non-negative quantities, so among these only the all-positive solution defines a Heronian triangle.

We could also conceivably mention that every positive-real-valued solution of this equation necessarily satisfies the triangle inequality (if a, b, c don't satisfy the triangle inequality then $$A^2$$ is negative). That is also a nontrivial and kind of interesting insight. D.Lazard's edit summary claimed The fundamental fact that the side lengths must satisfy the triangle inequality must not be replaced by trivias, but the triangle inequality was not mentioned in either footnote version, so I’m not sure what that comment is getting at.

As I said, I otherwise don’t understand why this footnote is helpful at all. –jacobolus (t) 19:05, 4 December 2022 (UTC)


 * The footnote is placed after a theorem statement and is required for WP:VERIFIABILITY. As, just now, I have not a reference under hands, a sketched proof allows verifiability, per WP:CALC.
 * In a recent edit, you replaced the condition that the factors of the right-hand side must be positive by the condition that the side lengths and the area are all positive. This makes the theorem more difficult to prove, as one has to show that, if $a, b, c, A$ are all positive, at most one of the factors of the right-hand side is not positive, and this implies that the equation becomes an equality between a positive square and non-positive product. So I'll revert your edit.
 * Also, your footnote is not useful since the non-positive solutions are not interesting in this article. It is also wrong, as the solution $$(A,a,b,c)=(0,1,2,3)$$ provides only eight 4-tuples instead of 16. D.Lazard (talk) 09:25, 5 December 2022 (UTC)
 * Aha. I just now figured out what your footnote was trying to say. I thought you were talking about positiveness of a, b, c, but you are talking about positiveness of (a + b + c), (-a + b + c), (a - b + c), and (a + b - c) being equivalent to a, b, c satisfying the triangle inequality. This needs to be made much more explicit if you want to bring readers along. I'll try to rewrite. –jacobolus (t) 16:43, 5 December 2022 (UTC)
 * Okay, I see you clarified it yourself but I gave rewriting this another shot. But feel free to keep working at it. I'm glad we had a conversation about this, because the footnote was previously confusing but I think we'll come out of this process with something that even dense readers like myself can follow. :-) –jacobolus (t) 17:16, 5 December 2022 (UTC)

can we break up the "properties" section into several separate sections, and rewrite the list as paragraphs?
This giant undifferentiated grab bag that turns every "property" into one bullet point seems like a poor structure for an encyclopedia article. Is it possible to group those bullets into a few groups and consolidate each one in its own section or sub-section with a heading, possibly a figure, etc.? Rewriting each bullet as ordinary paragraphs makes it much easier to reorganize or expand in the future with proofs, side comments, etc. Does anyone have any ideas about useful sections to include? –jacobolus (t) 21:25, 4 December 2022 (UTC)


 * Some possible sections: "Side lengths and perimeter", "Angles", "Altitudes and area", "Circumcircle", "Incircle and excircles", "Placement in a square lattice", ...? –jacobolus (t) 22:24, 4 December 2022 (UTC)

Using a triangle inscribed in a unit-diameter circle
Hi D.Lazard. I modified the half-angle tangent section to just use a triangle whose circumcircle has unit diameter so that the side lengths are equal to the sines of the opposite angles, and skip an intermediate scaling where we write the sides all as polynomials (rather than rational functions) of half-angle tangents. I felt like leaving out one level of indirection and associated re-assignment of the names $a$, $b$, $c$ would be clearer to follow and not lose anything. Does that seem reasonable, or are there reasons I am missing for keeping the intermediate step? The choice of scale to put a triangle in a unit-diameter circumcircle is very natural for trigonometry, cf. Kocik & Solecki (2009) "Disentangling a Triangle", which I should probably add as a reference here.)

Also I noticed that the Brahmagupta parametrization is almost but not quite related as previously described: the half-angle tangent parametrization modified as described instead gives the "c-extraversion" of Brahmagupta's Heronian triangle, to use John Conway's concept. Hopefully my footnote to that effect is clear enough for anyone who cares enough about the details to think about it. (I gave a reference to a paper where extraversion is defined/used, but it's possible there's a better canonical source. We should probably add a Wikipedia article about extraversion sometime in Conway's honor.) –jacobolus (t) 09:04, 31 December 2022 (UTC)
 * I am not opposed to most of your changes, as the best level of details is a question of taste, and my choices are not necessary the best ones.
 * Choosing a unit radius circumcircle would get rid of the factor 2 in the rational parametrization. I have not done this change because this is not fundamental, I have not checked the citation, and this would imply to change all scaling factors. Be free for doing it your self if you think this better.
 * I think that "c-extraversion" is too technical. Instead, I have introduced in the article a third way for clearing denominators, which gives exactly Brahmagupta's parametrization. I have also added some explanations on the meaning of the choices. D.Lazard (talk) 11:09, 31 December 2022 (UTC)
 * Choosing a unit-radius circumcircle would make everything twice as big and introduce an additional factor of 2. :-)
 * To get rid of the factor of 2 you need to use a circle of radius 1/4. But using a unit-diameter circumcircle is nice because then the sines of the angles are the same as the side lengths. –jacobolus (t) 11:16, 31 December 2022 (UTC)
 * Skipping extraversion is probably better. But I still want to add a Wikipedia article for it sometime. –jacobolus (t) 11:18, 31 December 2022 (UTC)

Does anyone have a link to an English translation of Brahmagupta?
The relevant part should be in the Brāhmasphuṭasiddhānta somewhere. The Carmichael book which was cited here does not mention Brahmagupta in the cited pages, and the Kurz paper does not mention where he found Brahmagupta's parametrization or explicitly support any claims about precisely what Brahmagupta did or did not prove about all Heronian triangles being included. Is there any better historical overview anywhere? Dickson's book is just a list of claims and references without much effort to put different works in context or make any narrative structure. –jacobolus (t) 19:59, 2 January 2023 (UTC)

According to Satya Prakash and also to Dickson, Brahmagupta's parametrization for oblique rational triangles was:
 * $$\begin{align}

a &= \tfrac12\Bigl(\frac{m^2}p + p\Bigr), \\ b &= \tfrac12\Bigl(\frac{m^2}q + q\Bigr), \\ c &= \tfrac12\Bigl(\frac{m^2}p - p\Bigr) + \tfrac12\Bigl(\frac{m^2}q - q\Bigr). \end{align}$$

... which is different than (a scaled version of) what we have written here, and based on rational rather than integer parameters. But also gives a good hint about how it was derived: namely by putting two right triangles with a common leg $$m$$ next to each-other, the sides of which are (rational) Pythagorean triples generated from Euclid's formula. It seems pretty obvious that this will generate all similarity classes of rational triangles given arbitrary rational numbers $$m, p, q.$$ –jacobolus (t) 20:18, 2 January 2023 (UTC)

Here is Colebrooke's (1817) translation:

"The square of an assumed quantity being twice set down, and divided by two other assumed quantities, and the quotients being severally added to the quantity first put, the moieties of the sums are the sides of a scalene triangle: from the same quotients the two assumed quantities being subtracted, the sum of the moieties of the differences is the base."

–jacobolus (t) 21:36, 2 January 2023 (UTC)