Permutation property of "law of cosines" stated more compactly through abstract algebra The law of cosines, stated: $c^2 = a^2 + b^2 - |a||b|\cos(\gamma)$ is a result that still holds if we swap the side-lengths and the corresponding angle.
Is there a way to capture this symmetry from a group-theoretic perspective? Or some other abstract algebraic perspective?
I apologize, I'm not entirely sure how to ask this due to significant lack of knowledge. I think one could use Galois theory to state this more elegantly -- by defining this to be a quadratic polynomial in some sense. Or we could not use Galois theory at all and think about a corresponding permutation group.
I'm also unaware if this is already a fairly common sidenote in the study of, say, quadratic forms for a particular geometry.
Motivation: Wherever I see this elementary result stated, either the other 2 corresponding equations are stated alongside it, or it's stated that the equation would still work if we swapped the variables. Having learned aspects of group theory and abstract algebra, I feel there is a better way to state this. And I'm hoping this more abstract way of interpreting this could lead to an elementary example of a more advanced concept -- and as a bonus, potentially open the door to better understanding applications of quadratic forms and of defining a geometry through a group-theoretic perspective.

Not sure what you want to do here, but if you're familiar with $S_3$, the symmetric group on $3$ letters,  you could probably use that here.  Each element of $S_3$, there are $6$, would permute $a,b$ and $c$ (let's say the angle  $\gamma $ changes automatically), and give a different version.
The elements are $\{e,(ab),(ac),(bc),(abc),(acb)\}$.
So $e$ does nothing.
Take $(ab)$.  It just interchanges $a$ and $b$.  So $c^2=a^2+b^2-\mid a\mid\mid b\mid\cos(\gamma) $ becomes $c^2=b^2+a^2-\mid b\mid\mid a\mid\cos(\gamma) $, which is the same statement.
We expect $3$ of them to be repeats, right?   Because there are only $3$ cases of the law of cosines.
But look at what $(ac)$ does.  We get $a^2=c^2+b^2-\mid c\mid\mid b\mid\cos(\alpha)$, where of course $\alpha$ is the angle opposite side $a$.
What do you think?
If you want a compact way of saying it, you could probably get away with saying that you can "cyclically permute" $a,b$ and $c$ (but of course you have to remember to change the angle), to get the other cases.
