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== Heuristic strategy to get Heron's formula ==

The formula for the area of a triangle (with sides lengths {{math|''a''}}, {{math|''b''}} and {{math|''c''}}):

* must be ''invariant under permutation of sides lengths'' {{math|''a''}}, {{math|''b''}} and {{math|''c''}};

* '''assuming the area formula is a polynomial''', it would have to be a [[homogeneous polynomial]] of degree 2 (since multiplying all side lengths by {{math|''m''}}, the area must be multiplied by {{math|''m''<sup>2</sup>}});

* for [[Degeneracy (mathematics)#Triangle|degenerate triangles]], i.e. when a side length equals the sum of the other 2 sides lengths, the area must be {{math|0}}, so the polynomial would have to be divisible by the product of the three [[trinomials]] {{math|(&minus; ''a'' + ''b'' + ''c'') (''a'' &minus; ''b'' + ''c'') (''a'' + ''b'' &minus; ''c'')}}: unfortunately this is a homogeneous polynomial of degree 3;  

* '''assuming the formula for "the square of the area of the triangle divided by the perimeter of the triangle" is a polynomial''', a homogeneous polynomial of degree 3 is just what we would need, i.e. {{math|{{sfrac|''A''<sup>2</sup>|(''a'' + ''b'' + ''c'')}} {{=}} ''k'' (&minus;''a'' + ''b'' + ''c'') (''a'' &minus; ''b'' + ''c'') (''a'' + ''b'' &minus; ''c'')}} for some positive constant {{math|''k''}};

* using the [[Pythagorean theorem]] for a right triangle with 2 orthogonal sides of length {{math|1}}, and thus hypothenuse {{math|{{sqrt|2}}}} and area {{math|{{sfrac|1|2}}}}, we have {{math|({{sfrac|1|2}})<sup>2</sup> {{=}} ''k'' (1 + 1 + {{sqrt|2}}) (&minus;1 + 1 + {{sqrt|2}}) (1 &minus; 1 + {{sqrt|2}}) (1 + 1 &minus; {{sqrt|2}})}} which yields {{math|''k'' {{=}} {{sfrac|1|16}}}}.

Thus, '''assuming the formula for "the square of the area of the triangle divided by the perimeter of the triangle" is a polynomial''', the [[Heuristic argument|heuristic]] suggests: {{math|''A'' {{=}} {{sfrac|1|4}} {{sqrt|(''a'' + ''b'' + ''c'') (&minus;''a'' + ''b'' + ''c'') (''a'' &minus; ''b'' + ''c'') (''a'' + ''b'' &minus; ''c'')}}}}.

Now, how can we turn this heuristic strategy into a proof?

A homogeneous trivariate (in {{math|''a''}}, {{math|''b''}} and {{math|''c''}}) homogeneous polynomial of degree 3 is of the form (with 10 coefficients)
: {{math|&alpha;<sub>3,0,0</sub> ''a''<sup>3</sup> + &alpha;<sub>2,1,0</sub> ''a''<sup>2</sup> ''b'' + &alpha;<sub>2,0,1</sub> ''a''<sup>2</sup> ''c'' + &alpha;<sub>1,2,0</sub> ''a'' ''b''<sup>2</sup> + &alpha;<sub>1,1,1</sub> ''a'' ''b'' ''c'' + &alpha;<sub>1,0,2</sub> ''a'' ''c''<sup>2</sup> + &alpha;<sub>0,3,0</sub> ''b''<sup>3</sup> + &alpha;<sub>0,2,1</sub> ''b''<sup>2</sup> ''c'' + &alpha;<sub>0,1,2</sub> ''b'' ''c''<sup>2</sup> + &alpha;<sub>0,0,3</sub> ''c''<sup>3</sup>.}}

'''Assuming the formula for "the square of the area of the triangle divided by the perimeter of the triangle" is a polynomial''', with 10 pairwise non-similar triangles for which we know the sides and areas, we can uniquely determine the above coefficients, to get  

: {{math|&minus; ''a''<sup>3</sup> + ''a''<sup>2</sup> ''b'' + ''a''<sup>2</sup> ''c'' + ''a'' ''b''<sup>2</sup> &minus; 2 ''a'' ''b'' ''c'' + ''a'' ''c''<sup>2</sup> &minus; ''b''<sup>3</sup> + ''b''<sup>2</sup> ''c'' + ''b'' ''c''<sup>2</sup> &minus; ''c''<sup>3</sup> {{=}} (&minus;''a'' + ''b'' + ''c'') (''a'' &minus; ''b'' + ''c'') (''a'' + ''b'' &minus; ''c'').}}

Now, how do we prove that

: '''Assuming the formula for "the square of the area of the triangle divided by the perimeter of the triangle" is a polynomial'''

is a valid assumption?

<!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/142.58.132.230|142.58.132.230]] ([[User talk:142.58.132.230#top|talk]]) 02:56, 21 February 2019 (UTC)</small>

: ''(Now correctly using the reply box.)'' In the heuristic, why is the following not tried, for some {{math|''k''}} to be determined?

:: {{math|''A''<sup>3</sup> {{=}} ''k'' [(&minus; ''a'' + ''b'' + ''c'') (''a'' &minus; ''b'' + ''c'') (''a'' + ''b'' &minus; ''c'')]<sup>2</sup>}} &mdash; [[User:Tentacles|Tentacles]]<sup>[[User_talk:Tentacles|Talk]] or <span style="font-size: 125%;">✉</span> [[Special:EmailUser/Tentacles|<tt>mailto:</tt>''Tentacles'']]</sup> 18:21, 21 January 2023 (UTC)

==Other Proofs==

There are other elegant proofs such as the one using coordinate geometry, starting out with the fact that half the cross product of two vectors is the area of the triangle they form, see the following:- https://www.quora.com/How-can-you-prove-Herons-formula-by-using-coordinate-geometry

There’s also several proofs such as the one using similar triangles and an incircle of unit radius and another using an incircle and an excircle and similar triangles and another using the double angle tangent identity (Euler’s) and another using an identity linking tan and cos, and Heron’s original proof and also a link to a proof using complex numbers, in the following link:-https://www.quora.com/Is-there-any-easy-way-to-prove-Herons-Theorem [[User:Overlordnat1|Overlordnat1]] ([[User talk:Overlordnat1|talk]]) 23:35, 4 May 2021 (UTC)

== In Heron's ''Metrica'' ==

Here’s a link to the original Greek (and German translation) https://archive.org/details/heronisalexandri03hero/page/19/mode/2up and here is an English translation https://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/HeroAlexandrinus/Metrica.i.1-9/Metrica.I.1-9.html#prop8 –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 20:41, 2 December 2022 (UTC)

== As a geometric mean ==

Let

:  {{math|''&alpha;'' {{=}} (1/2)(&minus; ''a'' + ''b'' + ''c''), ''&beta;'' {{=}} (1/2) (''a'' &minus; ''b'' + ''c''), ''&gamma;'' {{=}} (1/2) (''a'' + ''b'' &minus; ''c''),}}

then

: {{math|''A'' {{=}} {{sqrt|(''&alpha;''  ''&beta;'' ''&gamma;'') (''&alpha;'' + ''&beta;'' + ''&gamma;'')}} }}

so that the area is the geometric mean of the product and the sum of same three variables. &mdash; [[User:Tentacles|Tentacles]]<sup>[[User_talk:Tentacles|Talk]] or <span style="font-size: 125%;">✉</span> [[Special:EmailUser/Tentacles|<tt>mailto:</tt>''Tentacles'']]</sup> 03:22, 21 January 2023 (UTC)

:Welcome back to Wikipedia after a long absence from editing.
:Do you have another problem/topic where this point of view was insightful? It’s plausible one could be found, but there are plenty of other related formulas where these variables show up as <math>\alpha\beta(\alpha+\beta+\gamma)\big/\gamma,</math> <math>\alpha\beta\gamma \big/ (\alpha+\beta+\gamma)</math> etc. etc., where a geometric mean interpretation may not make much sense.
:An alternative point of view that I have seen in at least a couple sources before and seems fruitful is to take <math>s, s_a = s - a, s_b, s_c</math> as four variables such that <math>s_a+s_b+s_c -s = 0</math> (or for symmetry negate <math>s</math> or negate the other three), and then look what triangles and other objects are generated under various permutations and sign flips of those quantities. But I think this rather belongs in [[Heron's formula]] or other trigonometry articles rather than here per se. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 04:16, 21 January 2023 (UTC)

== Imaginary area ==

I deleted the following unsourced claim:
* If values are given such that a, b, and c do not correspond to a triangle, the value for A is [[Imaginary number|imaginary]].
While the sqrt in the Heron formula may be interpreted as an imaginary number if the argument is negative, that much is not notable. If the area is interpreted as imaginary, it should be sourced. [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 20:03, 16 January 2025 (UTC)

:The result of Heron's formula is imaginary. Interpreting this as an "area" is going to require some creativity and squinting. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 20:16, 16 January 2025 (UTC)
:Here's a source, a book review {{jstor|27642242}} (see also {{jstor|2695295}}, which won the [[Lester R. Ford Award]] in 2001)
:{{tqb|If Heron's formula is applied to find the area of a nonexistent triangle with sides {{tmath|1= a = 3}}, {{tmath|1= b = 1}}, and {{tmath|1= c = 1}}, the answer is {{tmath|1=\textstyle A = \tfrac14\sqrt{-45} \approx 1.677i}}. Despite appearances, this outwardly nonsensical result is meaningful. In fact, the triangle exists in {{tmath|\C^2}}, and Heron's formula correctly finds its (imaginary) area.}} A geometrical interpretation from these papers is as a triangle in a [[pseudo-Euclidean space|pseudo-Euclidean plane]] with [[metric signature|signature]] (1, 1). –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 20:45, 16 January 2025 (UTC)
::Excellent, thanks! [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 22:17, 16 January 2025 (UTC)
::I added content based on the ref, please review. [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 22:33, 16 January 2025 (UTC)
:::I think this would be better placed in a section toward the bottom rather than in the leading {{alink|Example}}. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 00:41, 17 January 2025 (UTC)
::::I moved it to the Generalizations section. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:51, 17 January 2025 (UTC)

== Notes section ==

@[[User:Jacobolus|Jacobolus]] (in response to {{diff|oldid=1269859493}}): It's definitely reasonable from a reader's perspective to not have a separate notes section for a single footnote, but I personally think that it would make the page more maintainable. Of course, Wikipedia's [[WP:NOTABOUTUS]], but I'm just curious if you agree or not. [[User:Gracen|/home/gracen/]] ([[User talk:Gracen|they]]/[[Special:Contributions/Gracen|them]]) 20:38, 17 January 2025 (UTC)

:I think it can work either to have separated or consolidated sections. Personally I prefer consolidated notes/references sections because they are simpler and because they forestall the sometimes occurring pedantic arguments about whether it's acceptable to add citations to text notes or add textual annotations to reference notes; instead any kind of note can include either type of material or both without worry. As a minor point I also prefer footnotes that say {{sup|[1]}} or {{sup|[a]}} instead of {{sup|[Note 1]}}. In this particular case where there's only one text note, I also don't think it's worth splitting it into a separate section. YMMV. If several editors prefer split sections I'm willing to abide by general preference though. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 00:39, 18 January 2025 (UTC)
::That all makes sense; thanks for indulging my curiosity. The main reason I think it's more maintainable is because I'm kind of an organization freak and I find it easier to work with material when it's all partitioned off into sections. Of course, it's just a matter of personal preference and I'm also willing to abide by consensus. [[User:Gracen|/home/gracen/]] ([[User talk:Gracen|they]]/[[Special:Contributions/Gracen|them]]) 14:39, 18 January 2025 (UTC)
:::In my opinion, the proper organization of articles places only references in the footnotes. Text, other than explanations of the relationship of sources to the content, belongs in the body of the article. Content in a footnotes is more difficult for readers to read and for editors to edit. Content in footnotes is not notable: if the content were notable it would appear in the article. Content in footnotes signals that the editor is forking the logic of the paragraph, almost never for a good reason. In my experience with Wikipedia footnotes, they often contradict the article, are original research or are trivia. Not 100%, but close.
:::In the case of this article, I think the footnote about the calculator belongs in a caption for the calculator widget. As it stands, the calculator is in a section with a formula, implying that is the formula is computed in the calculator. Only by reading the footnote do reader realize that it computes something different. (Yes, my bad). [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 16:30, 18 January 2025 (UTC)
::::It ''is'' the formula computed in the calculator, just computed in a specific order (which would make no difference whatsoever in any number system following the ordinary laws of arithmetic). Being careful to avoid numerical issues is essential for any computer implementation in inexact arithmetic, but the details here are irrelevant to most readers and should not be shoved in anyone's face. {{pb}} As for your more general point: I often feel the opposite. Many articles shove irrelevant trivia or technical minutiae into the main article body that should be put into footnotes instead (if only because when such information gets removed entirely it will inevitably be restored by insistent pedants). Being outside of the narrative flow and thus harder to read is the entire point of footnotes, and is beneficial whenever the content of the note would be distracting to the main point and flow of the article. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 16:56, 18 January 2025 (UTC)
:::::I think that in an article having short footnotes (which can contain explanations of the footnotes) and a separate bibliography section, explanatory footnotes and short footnotes possibly containing explanations of the footnotes should be consolidated into one section, not separate. In this case we have long footnotes instead, with no explanations, so separating them or consolidating them are both ok, but I prefer consolidated. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 17:59, 18 January 2025 (UTC)
:::::I agree that a footnote is a good place for the implementation detail. {{pb}} I don't have strong opinions about consolidating versus separating in this case. Leaving them consolidated seems fine. Having a "Notes" section for just one note seems a little unwieldy. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 19:09, 18 January 2025 (UTC)
:::::I completely agree about your second point. I see reference footnotes as being a replacement for inline text that says "this information is taken from <samp>[source]</samp>". Such information most certainly has a place in an encyclopedia (as does some trivia, as long as it has real value in inclusion), but just not in the regular flow of the text.{{pb}}As for the consolidation or separation of footnotes, I appreciate everyone who chimed in, and I wanted to share a benefit that I've noticed as a reader to grouping footnotes. My eyes usually skip over footnotes in the <sup>[1]</sup> style, but they don't skip over those in the <sup>[a]</sup> or <sup>[note 1]</sup> style. I think it's useful to indicate to readers what kind of footnote they're looking at so they can decide whether or not it's what they're looking for. If you're looking to verify a claim, you don't want to see an explanatory footnote, and if you're looking for further explanation you don't want a citation. [[User:Gracen|/home/gracen/]] ([[User talk:Gracen|they]]/[[Special:Contributions/Gracen|them]]) 20:27, 18 January 2025 (UTC)
::::::Purely explanatory footnotes are often better omitted or integrated into the main text. For a horrific example of explanatory footnotes gone bad, see [[24-cell]]. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 20:57, 18 January 2025 (UTC)
:::::::I disagree that purely explanatory footnotes should generally be omitted. Obviously [[24-cell]] is an example where they should, but I believe that there are quite a few less egregious cases where there's material of encyclopedic value that should be included but, unless you're an incredibly clever writer, you can't include neatly in the body text. I think a good example of this would be noting an degenerate case, such as the case of Heron's formula producing 0 for a degenerate triangle. Material such as that would distract from the main point of the text (that being that Heron's formula produces the area of an arbitrary triangle) but is clearly of encyclopedic value. [[User:Gracen|/home/gracen/]] ([[User talk:Gracen|they]]/[[Special:Contributions/Gracen|them]]) 20:18, 21 January 2025 (UTC)
::::::::The specific mention of degenerate triangles belongs in the main body, either in passing with a link to an article about [[degenerate triangle]]s (which should probably be expanded as a separate topic instead of just mentioned on a page about degeneracy in general), or if we want to say more than fits in the narrative of early sections in a section further down the page. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 21:06, 21 January 2025 (UTC)
::{{outdent|6}} In my opinion both could work, but hopefully the fact that I gave a poor example didn't detract from my point that certain details belong in explanatory footnotes, but that they should still be used with discretion. [[User:Gracen|/home/gracen/]] ([[User talk:Gracen|they]]/[[Special:Contributions/Gracen|them]]) 21:33, 21 January 2025 (UTC)
:::Yes, I agree that text notes are often valuable. A small handful of the notes at [[24-cell]] might be even worth keeping (but not the current 100 or whatever). Also often helpful are mixed text/reference notes where the text quotes, gives context for, or explains the reference, or where the text note is the main point but a reference supports the content of the note. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 22:52, 21 January 2025 (UTC)
::::Thanks, I appreciate it. I'm still relatively new here so I'm always willing to learn how to write better articles. [[User:Gracen|/home/gracen/]] ([[User talk:Gracen|they]]/[[Special:Contributions/Gracen|them]]) 16:00, 22 January 2025 (UTC)