p

r_{a}

r_{b}

r_{c}

r_{a}+r_{b}+r_{c}\geqslant p\sqrt{3}.

r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}=p^{2}

\frac{r_{a}^{2}+r_{b}^{2}}{2}\geqslant r_{a}r_{b},~\frac{r_{a}^{2}+r_{c}^{2}}{2}\geqslant r_{a}r_{c},~\frac{r_{b}^{2}+r_{c}^{2}}{2}\geqslant r_{b}r_{c},

r_{a}^{2}+r_{b}^{2}+r_{c}^{2}\geqslant r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}.

(r_{a}+r_{b}+r_{c})^{2}=r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+2(r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c})\geqslant3(r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c})=3p^{2}

r_{a}+r_{b}+r_{c}\geqslant p\sqrt{3}.

a=\frac{r_{a}(r_{b}+r_{c})}{\sqrt{r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}}},~b=\frac{r_{b}(r_{a}+r_{c})}{\sqrt{r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}}},~c=\frac{r_{c}(r_{a}+r_{b})}{\sqrt{r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}}}.

r_{a}+r_{b}+r_{c}\geqslant p\sqrt{3}~\Leftrightarrow~r_{a}+r_{b}+r_{c}\geqslant\frac{\sqrt{3}}{2}(a+b+c)~\Leftrightarrow~

~\Leftrightarrow~2(r_{a}+r_{b}+r_{c})\geqslant\sqrt{3}\left(\frac{r_{a}(r_{b}+r_{c})}{\sqrt{r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}}}+\frac{r_{b}(r_{a}+r_{c})}{\sqrt{r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}}}+\frac{r_{c}(r_{a}+r_{b})}{\sqrt{r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}}}\right)~\Leftrightarrow~

~\Leftrightarrow~2(r_{a}+r_{b}+r_{c})\geqslant\frac{2\sqrt{3}(r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c})}{\sqrt{r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}}}~\Leftrightarrow~

~\Leftrightarrow~(r_{a}+r_{b}+r_{c})^{2}\geqslant3(r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c})~\Leftrightarrow~r_{a}^{2}+r_{b}^{2}+r_{c}^{2}\geqslant r_{a}r_{b}+r_{a}r_{c}+r_{b}r_{c}~\Leftrightarrow~

~\Leftrightarrow~2r_{a}^{2}+2r_{b}^{2}+2r_{c}^{2}\geqslant2r_{a}r_{b}+2r_{a}r_{c}+2r_{b}r_{c}~\Leftrightarrow~(r_{a}-r_{b})^{2}+(r_{a}-r_{c})^{2}+(r_{b}-r_{c})^{2}\geqslant0.