# A number characterizing the deviation of a triangle from the regular triangle

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?

• 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. – bathalf15320 Apr 8 at 6:57
• @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. – Taras Banakh Apr 8 at 7:11
• Yes. I was tacitly normalising so that the longest side has length $1$--should have made that explicit. – bathalf15320 Apr 8 at 13:37
• MSE is a right forum for such type questions. – user64494 Apr 8 at 16:50
• @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. – Taras Banakh Apr 8 at 17:04

## 2 Answers

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.

• Note that $\dfrac{4A}{a^2+b^2+c^2}$ is the tangent of the Brocard angle $\omega$. Thus, $q = \left(1-\tan^2\omega\right)/2$. – darij grinberg Apr 8 at 18:34
• @darijgrinberg This should be an (perhaps the) answer, I think. – მამუკა ჯიბლაძე Apr 8 at 19:43
• I believe the book mentioned in the answer is idiomatic. See, for example dictionary.cambridge.org/us/dictionary/english/in-my-book – asahay Apr 8 at 22:48
• @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. – Taras Banakh Apr 9 at 4:35
• @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. – Taras Banakh Apr 9 at 4:39

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.

• 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. – Taras Banakh Apr 8 at 17:19