ILL 1989 Problem 28

In a triangle $ABC$ for which $6(a+b+c)r^2 = abc$ holds and where $r$ denotes the inradius of $ABC,$ we consider a point M on the inscribed circle and the projections $D,E, F$ of $M$ on the sides $BC=a, AC=b,$ and $AB=c$ respectively. Let $S, S_1$ denote the areas of the triangles $ABC$ and $DEF$ respectively. Find the maximum and minimum values of the quotient $\frac{S}{S_1}$