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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 :

$$ AB\cos(\widehat{C})+AC\cos(\widehat{B})+BC\cos(\widehat{A}) =\frac {A^2 + B^2 + C^2}{2} \tag{IV} $$

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    An attractively symmetrical formula!2011-07-03
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    @user6312: I agree. @Arjang: I'd prove this without appealing to the law of cosines. Indeed, in the usual trigonometric proof of the law of cosines you start by writing $c = a \cos{\beta} + b \cos{\alpha}$ and multiply this by $c$. Now do this for each side, add the three resulting equalities up and divide by two to get (IV). In order to get the law of cosines you write $a^2 + b^2 - c^2$ using these expressions and solve for $c^2$.2011-07-03
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    Oddly enough, this looks (to me) quite related to my answer here http://math.stackexchange.com/questions/38930/how-to-show-determinant-of-a-specific-matrix-is-nonnegative/49133#49133 (this would be the case for N=3, for N>3 we have the inequality)2011-07-03
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    @Theo : tx for the tip, I will be using it for revision.2011-07-05
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    Do you want to generalize $IV$ for quadrilaterals, pentagons, etc.?2011-08-02

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