Example 8. As shown in Figure 1, let the area $S$ of $\triangle ABC$ be 1. Try to find an internal point $D, E, F$ on each of the sides $BC, CA, AB$ respectively, such that the area $S'$ of $\triangle DEF$ satisfies $\frac{1}{4}<S'<\frac{1}{3}$. (1988, Jiangsu Junior High School Competition)
Let points $D, E, F$ be such that
$$
\frac{A F}{F B}=\frac{B D}{D C}=\frac{C E}{E A}=m .
$$

Then, by the theorem and the assumption $S=1$, we get
$$
S^{\prime}=\frac{m^{3}+1}{(m+1)^{3}}=\frac{m^{2}-m+1}{(m+1)^{2}} .
$$

From $\frac{1}{4}<\frac{m^{2}-m+1}{(m+1)^{2}}<\frac{1}{3}$, we solve to get
$$
\frac{1}{2}<m<2 \text { and } m \neq 1 \text {. }
$$

Therefore, for any $m$ satisfying $\frac{1}{2}<m<2$ and $m \neq 1$, by taking $D, E, F$ such that $\frac{A F}{F B}=\frac{B D}{D C}=\frac{C E}{E A}=m$, the area $S^{\prime}$ of $\triangle D E F$ must satisfy $\frac{1}{4}<S^{\prime}<\frac{1}{3}$.