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Maximum and minimum values of the quotient S/S_1

Source: IMO Longlist 1989, Problem 28

September 18, 2008
geometryinradiusgeometry unsolved

Problem Statement

In a triangle ABC ABC for which 6(a\plus{}b\plus{}c)r^2 \equal{} abc holds and where r r denotes the inradius of ABC, ABC, we consider a point M on the inscribed circle and the projections D,E,F D,E, F of M M on the sides BC\equal{}a, AC\equal{}b, and AB\equal{}c respectively. Let S,S1 S, S_1 denote the areas of the triangles ABC ABC and DEF DEF respectively. Find the maximum and minimum values of the quotient SS1 \frac{S}{S_1}
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