9. (10 points) In the figure, $ABCD$ is a parallelogram, $E$ is the midpoint of $CD$, the intersection of $AE$ and $BD$ is $F$, the intersection of $AC$ and $BE$ is $H$, the intersection of $AC$ and $BD$ is $G$, the area of quadrilateral $EHGF$ is 15 square centimeters, then the area of $ABCD$ is $\qquad$ square centimeters.
【Analysis】As shown in the figure, connect $E G$. According to the proportional relationship between the area of a triangle and its base, determine the relationship between $S_{\triangle B D E}$, $S_{\triangle D E F}$, $S_{\triangle B G H}$, and $S_{\text {quadrilateral } A B C D}$. Then, deduce the relationship between $S_{\text {quadrilateral } E H G F}$ and $S_{\text {quadrilateral }}$ $A B C D$. Finally, given that the area of quadrilateral $E H G F$ is 15 square centimeters, find the area of $A B C D$.

【Solution】Solution: As shown in the figure, connect $E G$. Since $E$ is the midpoint of $C D$,

so $D E=\frac{1}{2} C D$,
so $S_{\triangle B D E}=S_{\triangle A D E}=\frac{1}{4} S_{\text {quadrilateral } A B C D}$;
Since $A C$ and $B D$ intersect at $G$,
so $G$ is the midpoint of $A C$,
Since $E$ is the midpoint of $C D$,
so $E G / / A D$, and $\frac{\mathrm{EG}}{\mathrm{AD}}=\frac{1}{2}$,
so $\frac{\mathrm{EF}}{\mathrm{AF}}=\frac{\mathrm{EG}}{\mathrm{AD}}=\frac{1}{2}$,
so $S_{\triangle D E F}=\frac{1}{3} S_{\triangle A D E}=\frac{1}{12} S_{\text {quadrilateral } A B C D}$;
Since $E G / / A D$, and $A D / / B C$,
so $E G / / B C, \frac{\mathrm{EG}}{\mathrm{BC}}=\frac{1}{2}$,
so $\frac{\mathrm{GH}}{\mathrm{CH}}=\frac{\mathrm{EG}}{\mathrm{BC}}=\frac{1}{2}$,
so $S_{\triangle B G H}=\frac{1}{3} S_{\triangle B C G}=\frac{1}{12} S_{\text {quadrilateral } A B C D}$;
so $S_{\text {quadrilateral } E H G F}=S_{\triangle B D E}-S_{\triangle D E F}-S_{\triangle B G H}=\frac{1}{12} S_{\text {quadrilateral } A B C D}$,
so $S_{\text {quadrilateral } A B C D}=S_{\text {quadrilateral } E H G F} \times 12=15 \times 12=180$ (square centimeters)
Answer: The area of $A B C D$ is 180 square centimeters.
Therefore, the answer is: 180.