[ Equilateral (Regular) Triangle ]

In an equilateral triangle $D E F$, a circle of radius $r$ is inscribed. This circle is externally tangent to three other circles of the same radius at the points of tangency with the sides of the triangle. The centers of the external circles are $O_{1}, O_{2}, O_{3}$, respectively. Find the area of the hexagon formed by the intersection of triangles $D E F$ and $O_{1}, O_{2}, O_{3}$.
Prove that the circle inscribed in triangle $D E F$ is also inscribed in triangle $O_{1} O_{2} O_{3}$

## Answer

$2 r^{2} \sqrt{3}$