# 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