8. The centroid of $\triangle A B C$ is $G$, the midpoints of each side are $D, E, F, \overrightarrow{G D}+\overrightarrow{G E}+\overrightarrow{G F}=$
8. $\overrightarrow{0}$.

Let the midpoints of sides $B C$, $C A$, and $A B$ be $D$, $E$, and $F$ respectively, then 
$$
\begin{array}{l}
\overrightarrow{G D}+\overrightarrow{G E}+\overrightarrow{G F}=\frac{1}{3} \overrightarrow{A D}+\frac{1}{3} \overrightarrow{B E}+
\frac{1}{3} \overrightarrow{C F}=\frac{1}{3}(\overrightarrow{A B}+\overrightarrow{B D})+\frac{1}{3}(\overrightarrow{B C}+\overrightarrow{C E})+\frac{1}{3}(\overrightarrow{C A}+\overrightarrow{A F})=\frac{1}{3}(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A})+\frac{1}{3}(\overrightarrow{B D}+ \\
\overrightarrow{C D}+\overrightarrow{A F})=\overrightarrow{0} .
\end{array}
$$