9.8 (IMO 30 Pre) In $\triangle ABC$, it is known that $6(a+b+c) r^{2}=abc$. Consider a point $M$ on the incircle, and the projections of $M$ onto $BC$, $CA$, and $AB$ are $D$, $E$, and $F$, respectively. Let $S$ and $S_{1}$ denote the areas of $\triangle ABC$ and $\triangle DEF$, respectively. Find the maximum and minimum values of $\frac{S_{1}}{S}$.
9.8 From the formula $a b c=\frac{2 a}{\sin A} \cdot \frac{1}{2} b c \sin A=4 R S, \frac{r(a+b+c)}{2}=S$, substituting and simplifying yields: $R=3 r$. Also, from Euler's formula $I O^{2}=R(R-2 r)$, we get $I O=\sqrt{3} r>r$. The circumcenter is outside the incircle.

Let $I O$ intersect $\odot I$ at two points $K$ and $L$, with $K$ between $I$ and $O$. Then $O K \leqslant O M \leqslant O L<R$. Regarding the area formula of a triangle, we have: $S=2 R^{2} \cdot \sin A \sin B \sin C$; and Euler's formula for the area of the pedal triangle: $S_{1}=\left|R^{2}-O M^{2}\right| \cdot \frac{1}{2} \sin A \sin B \sin C$, $\therefore \frac{S_{1}}{S}=\frac{\mid R^{2}-O M^{2} \perp}{4 R^{2}}$. Thus, from the range of $O M$, we can deduce $\frac{5-2 \sqrt{3}}{36} \leqslant \frac{S_{1}}{S} \leqslant \frac{5+2 \sqrt{3}}{36}$.