Question

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

Answer 1

Distance between excenter $I_1$ and incenter $I$ is
$I{I}_1=4R\sin \frac{A}{2}=4$ (given)

Since triangle is equilateral, we have $4=4R\sin \frac{\pi }{6}$ or
$R=2$ or $R=2r=2$ or
$r=1$