10. As shown in Figure $2, P$ is a point inside $\triangle A B C$, and the extensions of $B P$, $C P$, and $A P$ intersect $A C$, $A B$, and $B C$ at points $E$, $F$, and $D$ respectively. Consider the following three equations:
(1) $\frac{S_{\triangle A B P}}{S_{\triangle A P C}}=\frac{B D}{C D}$;
(2) $\frac{S_{\triangle B P C}+S_{\triangle A P C}}{S_{\triangle B P C}}=\frac{A B}{B F}$;
(3) $\frac{C E}{E A} \cdot \frac{A B}{B F} \cdot \frac{F P}{P C}=1$.
Among these, the number of correct equations is ( ) .
(A) 0
(B) 1
(C) 2
(D) 3
10.D.

From $\frac{S_{\triangle A B P}}{S_{\triangle B P D}}=\frac{S_{\triangle A P C}}{S_{\triangle P C D}}=\frac{A P}{P D} \Rightarrow \frac{S_{\triangle A P B}}{S_{\triangle A P C}}=\frac{S_{\triangle B P D}}{S_{\triangle P C D}}=\frac{B D}{C D}$, therefore
(1) is correct;
From $\frac{A B}{B F}=\frac{S_{\triangle A B C}}{S_{\triangle B F C}}=\frac{S_{\triangle A B P}}{S_{\triangle B F P}}=\frac{S_{\triangle A B C}-S_{\triangle A B P}}{S_{\triangle B F C}-S_{\triangle B F P}}=$
$\frac{S_{\triangle B P C}+S_{\triangle A P C}}{S_{\triangle B P C}}$, therefore (2) is correct;
From $\frac{C E}{E A} \cdot \frac{A B}{B F} \cdot \frac{F P}{P C}=\frac{S_{\triangle B P C}}{S_{\triangle A B P}} \cdot \frac{S_{\triangle B P C}+S_{\triangle A P C}}{S_{\triangle B P C}}$.
$\frac{S_{\triangle A P B}}{S_{\triangle A P C}+S_{\triangle B P C}}=1$, therefore (3) is correct.