Triangles with Proportional Sides

Geometry Level 5

Let SS be the set of all triangles which have side lengths a,b,c a, b, c which satisfy the equation ab=bc \frac {a}{b} = \frac {b}{c}. For each triangle ABCABC in the set SS, let MABCM_{ABC} denote the value of ac\frac {a}{c}. Let RR be the smallest real number such that MABCR M_{ABC} \leq R for all triangles ABCABC in the set SS. The value of 2R 2R can be expressed as a+b a +\sqrt{b}, where aa and bb are positive integers. What is the value of a+ba+b?

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