What more can be said about the identity derived from law of cosines (motivation below)$$ \cos(\widehat{A})BC+ A\cos(\widehat{B})C+ AB\cos(\widehat{C})+=\frac {A^2 + B^2 + C^2}{2} \tag{IV}$$
RHS seems as if operator $\cos(\widehat{\phantom{X}})$ is being applied consecutively to terms of ABC, I tried to represent it in an analogous way to the Laplacian operator convention, but maybe there are more common ways of representing RHS using some operator and sigma notation ( please let me know if there is).
My question is : Are there any identities/structures relating or looking similar to IV, I apologize if this looks like a general fishing expedition question but I can not think of anything more that I can add to this post at this stage. Thank you
Motivation for IV,
Let $A,B,C$ be a triangle.
Let $\widehat{C} = \widehat{AB}$ stand for the Angle opposite to side C between the sides A and B, then the law of cosines for all three sides can be written as $$ A^2 + B^2 - 2 AB\cos(\widehat{C}) = C^2 \tag{I} $$ $$ A^2 + C^2 - 2 AC\cos(\widehat{B}) = B^2 \tag{II} $$ $$ B^2 + C^2 - 2 BC\cos(\widehat{A}) = A^2 \tag{III} $$ Adding $I ,II,III$ and juggling the terms we get :
$$ AB\cos(\widehat{C})+AC\cos(\widehat{B})+BC\cos(\widehat{A}) =\frac {A^2 + B^2 + C^2}{2} \tag{IV} $$