$\mathrm{acot}\mathrm{A}+\mathrm{bcot}\mathrm{B}+\mathrm{ccot}\mathrm{C}$ is equal to

$\begin{array}{l}a\cot A+b\cot B+c\cot C\\ =2R(\sin A\cot A+\sin B\cot B+\sin C\cot C]\\ =2R(\cos A+\cos B+\cos C)\\ =2R\left(1+4\sin \frac{1}{2}A\sin \frac{1}{2}B\sin \frac{1}{2}C\right)\\ =2R\left(1+\frac{r}{R}\right)=2(R+r).\end{array}$