If the distance between incenter and one of the excenter of an equilateral triangle is 4 units, then find the inradius of the triangle.

Explanation

To solve the problem, we need to find the inradius of an equilateral triangle given that the distance between the incenter (I) and one of the excenters (say, I_A) is 4 units.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
In an equilateral triangle, the distance between the incenter (I) and an excenter (I_A) can be expressed using the formula:
\[
d = \frac{r}{\sin\left(\frac{A}{2}\right)}
...