# Why excircles?

Geometry Level 5

Let $$I_1,I_2,I_3$$ be the centres of the excircles of $$\Delta ABC$$. Denote the area of $$\Delta ABC$$ by $$\Delta_O$$ and the area of $$\Delta I_1I_2I_3$$ by $$\Delta_I$$.

Define the set $$E$$ as follows: $E=\left\{k\in \mathbb{R}\:|\:\Delta_I\geq k\Delta_O\: \text{for all triangles}\:\Delta ABC\right\}$

Enter the value of $$\sup \{E\}$$ up to three decimal places.

Notations:

• $$\sup$$ denotes the supremum of a set.
• $$\mathbb{R}$$ denotes the set of all real numbers.

This problem is part of my set: Geometry

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