\(\overrightarrow{GM}+\overrightarrow{GN}+\overrightarrow{GP}=\overrightarrow{GA}+\overrightarrow{AM}+\overrightarrow{GB}+\overrightarrow{BN}+\overrightarrow{GC}+\overrightarrow{CP}\)
\(=\overrightarrow{AM}+\overrightarrow{BN}+\overrightarrow{CP}=\frac{2}{3}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{BC}+\frac{2}{3}\overrightarrow{CA}\)
\(=\frac{2}{3}\overrightarrow{CB}+\frac{1}{2}\overrightarrow{BC}=-\frac{1}{6}\overrightarrow{BC}\)