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



About the inscribed sphere and the exspheres of a $n$-dimensional simplex
Let $S\_1,\dots,S\_{n+1}$ be the $(n-1)$-dimensional volumes of the corresponding faces, and $V$ be the $n$-dimensional volume of the simplex. Then $\displaystyle V=\frac{r\_0S}n=\frac{r\_i(S-2S\_i)}n$, where $S=\sum\_i S\_i$. Thus 
$$
 \frac {n-1}{r\_0}=\frac {(n-1)S}{nV}=\frac{\sum\_i (S-2S\_i)}{nV}
 =\sum\_i\frac{1}{r\_i},
$$
as required.


