Example 2 As shown in Figure 1.13.2, the areas of the four small triangles are marked. Find the area of $\triangle A B C$.

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Let $D$ be the center of mass of particles $B, C$. Since $B D: D C=40: 30=4: 3$, we take $m_{B}=$ $3 a, m_{C}=4 a$. Let $P$ be the center of mass of the particle system $A, B, C$, then $P$ is also the center of mass of particles $B, E$, hence $S_{\triangle B P C}: S_{\triangle B E C}=B P:(B P+P E)=m_{E}:\left(m_{B}+m_{E}\right)$, solving this gives $m_{E}=6 a$. By property 3, we know that $E$ is the center of mass of particles $A, C$, then $m_{A}=m_{E}-m_{C}=2 a$, and $A E: E C=m_{C}: m_{A}=2: 1$.
Thus we have
$$
S_{\triangle A B C}=3 S_{\triangle B E C}=3 \cdot 105=405
$$