# Triangles with Proportional Sides

Geometry Level 5

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

×