<s>[INST] Proving that $a \cos A+b\cos B+c \cos C$ is the semi-perimeter of $\triangle ABC$ 
With each symbol having its usual meaning, prove that in $\triangle ABC$,
$$a \cos A+b\cos B+c \cos C=s$$
where $s$ is the semi-perimeter.


I started by applying the rule of sines:

*

*$a=k\sin A$

*$b=k\sin B$

*$c=k \sin C$
where $k$ is a constant

so, we have:

$k(\sin A\cos A+\sin B\cos B+\sin C\cos C)$

manipulating this,  we get:

$\frac{k}{2}(\sin 2A+\sin 2B+\sin 2C)$


$\frac{k}{2}(2\sin(A+B)\cos(A-B) + \sin 2C)$


$\frac{k}{2}(2\sin C\cos(A-B)+\sin 2C)$


$\frac{k}{2}2\sin C(\cos(A-B)+\cos C)$

But this doesn't seem to be yielding desired results. Could someone help me out with this? Or maybe tell me a better approach to this problem?
 [/INST] What is true is the relation $a\cos A+b \cos B+c\cos C=\dfrac{2rs}{R}$ where $r$ is the inradius, $R$ is the circumradius and $s$ is the semi perimeter of the triangle.
By the sine rule, we can write the length of the sides as $a=2R\sin A,b=2R\sin B,c=2R\sin C$ therefore we have $$a\cos A+b \cos B+c\cos C=R(\sin 2A+\sin 2B+\sin 2C)$$
It is well known that in a triangle, we have $\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B \sin C$
So we have $a\cos A+b \cos B+c\cos C=4R\sin A\sin B \sin C$
Again, by the sine rule, re writing $\sin A=\dfrac{a}{2R}$ and similarly for other sides, we obtain $a\cos A+b \cos B+c\cos C=\dfrac{abc}{2R^2}$
It is also well known that $abc$ and $\Delta$ are related by the relation $\Delta=\dfrac{abc}{4R}$, so substituting the value into the equation we obtain the value as $\dfrac{2\Delta}{R}$, but since $\Delta=rs$, we have $$a\cos A+b \cos B+c\cos C=\dfrac{2rs}{R}$$

Helpful: Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
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