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Given a triangle $\Delta$ with sides of length $a\le b\le c$, consider the number $$q=\frac{a^4+b^4+c^4}{(a^2+b^2+c^2)^2}$$ and observe that $\frac13\le q\le\frac12$ and the extremal values of $q$ characterize some geometric properties of the triangle $\Delta$. Namely:

$\bullet$ $q=\frac13$ if and only if $a=b=c$ (which means that the triangle $\Delta$ is regular);

$\bullet$ $q=\frac12$ if and only if $c=a+b$ (which means that the triangle $\Delta$ is degenerated).

I am writing a paper (in applications of math to Electric Engineering) where the number $q$ is applied for evaluation of the deviation of a triangle (describing the quality of 3-phase electric energy) from being regular, and need to call the number $q$ somewhow (for example, quadrofactror), but wonder if $q$ already has some standard name. This motivates my

Question. Has the number $q$ some standard name in Plane Geometry?

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  • $\begingroup$ This is not an answer but just a comment that there is another function of the side lengths which does what you want--its minimum (zero) is taken at the degenerate cases and its maximum at the equilateral one. It does have a name (area) and its expression as a function of $a$, $b$ and $c$ is Heron's formula. $\endgroup$
    – bathalf15320
    Commented Apr 8, 2021 at 6:57
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    $\begingroup$ @bathalf15320 Thank you for the comment. The area is a good function but it is not invariant under similarity transformations. So, it does not evaluate the form (and the regularity) of the triangle. $\endgroup$
    – Taras Banakh
    Commented Apr 8, 2021 at 7:11
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    $\begingroup$ Yes. I was tacitly normalising so that the longest side has length $1$--should have made that explicit. $\endgroup$
    – bathalf15320
    Commented Apr 8, 2021 at 13:37
  • $\begingroup$ MSE is a right forum for such type questions. $\endgroup$
    – user64494
    Commented Apr 8, 2021 at 16:50
  • $\begingroup$ @user64494 Probably you are right concerning MSE, but I am not a member of MSE and would not like to register there only in sake of this single question. $\endgroup$
    – Taras Banakh
    Commented Apr 8, 2021 at 17:04

2 Answers 2

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Added to my comment above, this time taking care of my carelessness in not normalising: one has the formula $$\frac{16 A^2}{(a^2+b^2+c^2)^2}=1-\frac{2(a^4+b^4+c^4)}{(a^2+b^2+c^2)^2} $$ which shows, at least in my book, that a normalised version of the area $A$ (more precisely of its square) does the trick.

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    $\begingroup$ Note that $\dfrac{4A}{a^2+b^2+c^2}$ is the tangent of the Brocard angle $\omega$. Thus, $q = \left(1-\tan^2\omega\right)/2$. $\endgroup$
    – darij grinberg
    Commented Apr 8, 2021 at 18:34
  • $\begingroup$ @darijgrinberg This should be an (perhaps the) answer, I think. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 8, 2021 at 19:43
  • $\begingroup$ I believe the book mentioned in the answer is idiomatic. See, for example dictionary.cambridge.org/us/dictionary/english/in-my-book $\endgroup$
    – Anurag Sahay
    Commented Apr 8, 2021 at 22:48
  • $\begingroup$ @asahay Ups! But at least this paper about Heron formula maa.org/sites/default/files/images/upload_library/22/Ford/… is quite real. I then rewrite what I wanted to say to user bathhalf5320. $\endgroup$
    – Taras Banakh
    Commented Apr 9, 2021 at 4:35
  • $\begingroup$ @bathhalf15320 Thank you very much for your answer. It was very helpful. In fact, my formulas involved $q$ in the subformula $\sqrt{3−6q}$ and now I understand what this subformula actually means: it is the normalized area of the triangle! Great! I would like to write acknowledgement to your help. Should I write your nick bathhalf15320 or some real name will be better? Thank you. $\endgroup$
    – Taras Banakh
    Commented Apr 9, 2021 at 4:39
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See "List of trianle inequalities" , especially its "Side lengths" section, to this end . There is a reference Posamentier, Alfred S. and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012., p. 261 there.

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    $\begingroup$ Indeed, in the mentioned paper on Wikipedia there is a similar inequality with the same lower and upper bounds: $\frac13\le \frac{a^2+b^2+c^2}{(a+b+c)^2}<\frac12$. But the upper bound $\frac12$ is attainable only at triangles with two coinciding vertices. On the other hand, the upper bound $\frac12$ at my inequality is attainable at each triangle on a line. $\endgroup$
    – Taras Banakh
    Commented Apr 8, 2021 at 17:19

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