a) Ta có: 

BC = $\sqrt{(R + R')^{2} - (R - R')^{2}}$ = $\sqrt{R^{2} + 2RR' + R'^{2} - R^{2} + 2RR' - 2R'^{2}}$ =  $\sqrt{4RR'}$ = 2$\sqrt{RR'}$

b) Ta có:

MB = $\sqrt{(R + r)^{2} - (R - r)^{2}}$ = 2$\sqrt{Rr}$

MC = $\sqrt{(R' + r)^{2} - (R' - r)^{2}}$ = 2$\sqrt{R'r}$

MB + MC = BC 

$\Rightarrow $ 2$\sqrt{Rr}$ + 2$\sqrt{R'r}$ = 2$\sqrt{RR'}$

$\Leftrightarrow $ $\sqrt{Rr}$ + $\sqrt{R'r}$ = $\sqrt{RR'}$

$\Leftrightarrow $ $\sqrt{r}$ = $\frac{\sqrt{RR'}}{\sqrt{R} + \sqrt{R'}}$ 

$\Rightarrow $ r = $\frac{RR'}{(\sqrt{R} + \sqrt{R'})^{2}}$.

Vậy r = $\frac{RR'}{(\sqrt{R} + \sqrt{R'})^{2}}$.