Example 3 Given that the area of a convex quadrilateral is $S$, take a point inside it and construct its symmetric points with respect to the midpoints of each side, obtaining the four vertices of a new quadrilateral. Find the area of the new quadrilateral.
As shown in Figure 3.3.11, let $O$ be a point inside the quadrilateral $A B C D$, and let $E, F, G, H$ be the midpoints of the sides, respectively. $E_{1}, F_{1}, G_{1}, H_{1}$ are the symmetric points of point $O$ with respect to these midpoints.

Since $E F$ is the midline of $\triangle E_{1} O F_{1}$, then $S_{\triangle E_{1} O F_{1}} = 4 S_{\triangle E O F}$. Similarly, we have $S_{\triangle F_{1} O G_{1}} = 4 S_{\triangle F O G}$, $S_{\triangle G_{1} O H_{1}} = 4 S_{\triangle C O H}$, and $S_{\triangle H_{1} O E_{1}} = 4 S_{\triangle H O E}$. Therefore, $S_{E_{1} F_{1} G_{1} H_{1}} = 4 S_{E F G H}$. By Theorem 6, hence $S_{E_{1} F_{1} G_{1} H_{1}} = 2 S$.