17.5 Given that the area of $\triangle A B C$ is $S$, points $D$, $E$, and $F$ are the midpoints of $B C$, $C A$, and $A B$ respectively, $I_{1}$, $I_{2}$, and $I_{3}$ are the incenters of $\triangle A E F$, $\triangle B F D$, and $\triangle C D E$ respectively, and the area of $\triangle I_{1} I_{2} I_{3}$ is $S^{\prime}$. Find the value of $S^{\prime} / S$.
17.5 Let the incenter of $\triangle D E F$ be $I$. It is easy to know that $\triangle A F E \cong \triangle F B D \cong \triangle E D C \cong \triangle D E F$. Therefore, their inradii are equal. And their areas are $S / 4$.

Let $I I_{1}$ intersect $E F$ at point $H$, then $\triangle A F E$ and $\triangle D E F$ are homothetic about point $H$, $\therefore H I_{1}=H I ; \triangle I_{1} H E \cong \triangle I H F$; $H E=H F$. Let $I I_{2}$ intersect $F D$ at point $K$, then $K I_{2}=K I . \therefore I_{1} I_{2}=2 G H=D E$, similarly, $\cdots \cdots$. Thus, $\triangle I_{1} I_{2} I_{3} \cong$ $\triangle D E F, S^{\prime}: S=1 / 4$