【Question 17】
As shown in the figure, the diagonals $A C$ and $B D$ of quadrilateral $A B C D$ intersect at point $O$. If $S_{\triangle A B D}=10, S_{\triangle A B C}=12, S_{\triangle B D C}=20$, what is the area of $S_{\triangle B O C}$?
【Analysis and Solution】
Equal area transformation, butterfly model of any quadrilateral.
Since $\frac{S_{\triangle A O B}}{S_{\triangle B O C}}=\frac{A O}{C O}, \frac{S_{\triangle A O D}}{S_{\triangle C O D}}=\frac{A O}{C O}$;
Therefore, $\frac{S_{\triangle A O B}+S_{\triangle A O D}}{S_{\triangle B O C}+S_{\triangle C O D}}=\frac{A O}{C O}$, which means $\frac{S_{\triangle A B D}}{S_{\triangle B D C}}=\frac{A O}{C O}$;
Since $S_{\triangle A B D}=10, S_{\triangle B D C}=20$;
Therefore, $\frac{A O}{C O}=\frac{10}{20}=\frac{1}{2}$;
Therefore, $\frac{S_{\triangle A O B}}{S_{\triangle B O C}}=\frac{1}{2}$;
Therefore, $S_{\triangle B O C}=S_{\triangle A B C} \times \frac{2}{3}=12 \times \frac{2}{3}=8$.