IMO 1989 LL GRE30

Origin: GRE

Problem

In a triangle ABC for which 6(a + b + c)r2 = abc, we consider a point M on the inscribed circle and the projections D, E, F of M on the sides BC, AC, and AB respectively. Let S, S1 denote the areas of the triangles ABC and DEF respectively. Find the maximum and minimum values of the quotient S S1 (here r denotes the inradius of ABC and, as usual, a = BC, b = AC, c = AB).