Let us consider $n$-dimensional simplex $K$ in $n$-dimensional Euclidean space. Let $r_0$ be the radius of the inscribed sphere of $K$, and let be $r_1, r_2, \cdots, r_{n+1}$ be each radius of the [exsphere][1] of $K$. 

Then, here is my question.

>**Question** : How can we represent $r_0$ by $r_1, r_2, \cdots, r_{n+1}$?

**Remark** : This question has been [asked previously on math.SE][2] without receiving any answers.

**Motivation** : I've known the followings : 

In the $n=2$ case, 
$$r_0=\frac{1}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}}.$$
In the $n=3$ case, 
$$r_0=\frac{2}{\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}}.$$

Then, I reached the following conjecture (of course, this is just a conjecture) : 
$$r_0=\frac{n-1}{\sum_{i=1}^{n+1}\frac{1}{r_i}}.$$

I don't have any good idea for $n$ in general. Can anyone help?


  [1]: http://en.wikipedia.org/wiki/Exsphere_%28polyhedra%29
  [2]: http://math.stackexchange.com/questions/535692/about-the-inscribed-sphere-and-the-escribed-spheres-of-a-n-dimensional-simplex