Inradius of Triangle in Terms of Exradii

Theorem
Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.

Then we have the following relation:
 * $\dfrac 1 r = \dfrac 1 {\rho_a} + \dfrac 1 {\rho_b} + \dfrac 1 {\rho_c}$

where:
 * $r$ is the inradius
 * $\rho_a$, $\rho_b$ and $\rho_c$ are the exradii of $\triangle ABC$ $a$, $b$ and $c$ respectively.

Proof
The area $\AA$ of $\triangle ABC$ is given by:

Similarly:

Hence: