# Malfatti Circles!

Geometry Level 5

Let $$ABC$$ be a triangle with in-radius $$r$$. Let $${ \Gamma }_{ 1 }, { \Gamma }_{ 2 }, { \Gamma }_{ 3 }$$ be three circles inscribed inside $$ABC$$ such that each touches other circles and also two of the sides. (Such a configuration is called Malfatti circles).

Let $${ O }_{ 1 }, { O }_{ 2 }, { O }_{ 3 }$$ be respectively the centres of the circles $${ \Gamma }_{ 1 }, { \Gamma }_{ 2 }, { \Gamma }_{ 3 }$$. If $$r'$$ denotes the in-radius of $${ O }_{ 1 }{ O }_{ 2 }{ O }_{ 3 }$$,

Find the minimum value of $$\frac r{r'}$$ to 3 decimal places.

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