Example 4 Given that the area of quadrilateral $A B C D$ is $9, P$ is a point inside the quadrilateral. Let the centroids of $\triangle A B P, \triangle B C P, \triangle C D P$, and $\triangle D A P$ be $G_{1}, G_{2}, G_{3}, G_{4}$ respectively. Find the area of quadrilateral $G_{1} G_{2} G_{3} G_{4}$.
Let $e=(\overrightarrow{A D} \times \overrightarrow{A B})^{0}$.
Then $18 S_{\text {quadrilateral } G_{1} G_{2} G_{3} G_{4}} e$
$$
\begin{array}{l}
=9\left(G_{3}-G_{1}\right) \times\left(G_{4}-G_{2}\right) \\
=(C+D-A-B) \times(D+A-B-C) \\
=2(C-A) \times(D-B) \\
=4 S_{\text {quadrilateral } A B C D} e .
\end{array}
$$

Therefore, $S_{\text {quadrilateral } G_{1} G_{2} G_{3} G_{4}}=\frac{2}{9} \times 9=2$.