Given three circumferences of radii $r$ , $r'$ and $r''$ , each tangent externally to the other two, calculate the radius of the circle inscribed in the triangle whose vertices are their three centers.
1. **Identify the sides of the triangle formed by the centers of the circles:**
   Given three circles with radii \( r \), \( s \), and \( t \), each tangent externally to the other two, the distances between the centers of these circles are:
   \[
   AB = r + s, \quad BC = s + t, \quad CA = t + r
   \]
   These distances form the sides of the triangle \( \triangle ABC \).

2. **Calculate the semi-perimeter of the triangle:**
   The semi-perimeter \( p \) of \( \triangle ABC \) is given by:
   \[
   p = \frac{(r+s) + (s+t) + (t+r)}{2} = \frac{2r + 2s + 2t}{2} = r + s + t
   \]

3. **Use Heron's formula to find the area of the triangle:**
   Heron's formula for the area \( \Delta \) of a triangle with sides \( a \), \( b \), and \( c \) and semi-perimeter \( p \) is:
   \[
   \Delta = \sqrt{p(p-a)(p-b)(p-c)}
   \]
   Substituting \( a = r+s \), \( b = s+t \), \( c = t+r \), and \( p = r+s+t \):
   \[
   \Delta = \sqrt{(r+s+t)(r+s+t-(r+s))(r+s+t-(s+t))(r+s+t-(t+r))}
   \]
   Simplifying the terms inside the square root:
   \[
   \Delta = \sqrt{(r+s+t)(t)(r)(s)} = \sqrt{rst(r+s+t)}
   \]

4. **Calculate the radius of the inscribed circle:**
   The radius \( r_{\text{in}} \) of the inscribed circle in a triangle is given by:
   \[
   r_{\text{in}} = \frac{\Delta}{p}
   \]
   Substituting \( \Delta = \sqrt{rst(r+s+t)} \) and \( p = r+s+t \):
   \[
   r_{\text{in}} = \frac{\sqrt{rst(r+s+t)}}{r+s+t} = \sqrt{\frac{rst}{r+s+t}}
   \]

The final answer is \(\boxed{\sqrt{\frac{rst}{r+s+t}}}\).