Relationships of Radii 2

Sorry about the delay; I was away for some days. First notice that: $s(s-b)(s-c)+s(s-c)(s-a)+s(s-a)(s-b)-(s-a)(s-b)(s-c) = abc$ where $s$ is the semi-perimeter. This is nicely seen using the identity: $(x+y+z)(xy+yz+zx) - xyz \equiv (x+y)(y+z)(z+x)$ and substituting $x=s-a$ etc.

Now $\mathrm{Area}^2 = s(s-a)(s-b)(s-c)$ by Heron's Formula, and $\mathrm{Area} = \frac{abc}{4R}$ .

So $\frac{\mathrm{Area}^2}{s-a} + \frac{\mathrm{Area}^2}{s-b} + \frac{\mathrm{Area}^2}{s-c} - \frac{\mathrm{Area}^2}{s} = \mathrm{Area} \times 4R$

And using the Area Formulae on the Incircles and Excircles Wiki , this gives us $r_1 + r_2 + r_3 -r = 4R$ as required.

Hope this helps!

Check the Wiki article