r

r_{a}

r_{b}

r_{c}

ABC

BC=a

AC=b

AB=c

I_{a}

I_{b}

I_{c}

R

ABC

p

r_{a}+r_{b}+r_{c}-r=4R

S_{\triangle I_{a}I_{b}I_{c}}=2pR

r=\frac{S_{\triangle ABC}}{p},~r_{a}=\frac{S_{\triangle ABC}}{p-a},~r_{b}=\frac{S_{\triangle ABC}}{p-b},~r_{c}=\frac{S_{\triangle ABC}}{p-c},

\frac{1}{r_{a}}+\frac{1}{r_{b}}+\frac{1}{r_{c}}=\frac{1}{r},~S_{\triangle ABC}=\frac{abc}{4R}=\sqrt{p(p-a)(p-b)(p-c)},

r_{a}+r_{b}+r_{c}-r=\frac{S_{\triangle ABC}}{p-a}+\frac{S_{\triangle ABC}}{p-b}+\frac{S_{\triangle ABC}}{p-c}-\frac{S_{\triangle ABC}}{p}=

=\frac{S_{\triangle ABC}\cdot abc}{p(p-a)(p-b)(p-c)}=\frac{S_{\triangle ABC}\cdot abc}{S^{2}_{\triangle ABC}}=\frac{abc}{S_{\triangle ABC}}=4R.

S_{\triangle I_{a}I_{b}I_{c}}=S_{\triangle I_{a}BC}+S_{\triangle I_{b}AC}+S_{\triangle I_{c}AB}+S_{\triangle ABC}=

=\frac{1}{2}ar_{a}+\frac{1}{2}br_{b}+\frac{1}{2}cr_{c}+pr=

=\frac{1}{2}ar_{a}+\frac{1}{2}pr_{a}-\frac{1}{2}pr_{a}+\frac{1}{2}br_{b}+\frac{1}{2}pr_{b}-\frac{1}{2}pr_{b}+\frac{1}{2}cr_{c}+\frac{1}{2}pr_{c}-\frac{1}{2}pr_{c}+pr=

=\frac{1}{2}r_{a}(a-p)+\frac{1}{2}r_{b}(b-p)+\frac{1}{2}r_{c}(p-c)+\frac{1}{2}p(r_{a}+r_{b}+r_{c})-\frac{1}{2}pr+\frac{3}{2}pr=

=\frac{1}{2}p(r_{a}+r_{b}+r_{c}-r)-\frac{1}{2}r_{a}(p-a)-\frac{1}{2}r_{b}(p-b)-\frac{1}{2}r_{c}(p-c)+\frac{3}{2}pr=

=\frac{1}{2}p(r_{a}+r_{b}+r_{c}-r)-\frac{1}{2}S_{\triangle ABC}-\frac{1}{2}S_{\triangle ABC}-\frac{1}{2}S_{\triangle ABC}+\frac{3}{2}pr=

=\frac{1}{2}p\cdot4R-\frac{3}{2}S_{\triangle ABC}+\frac{3}{2}S_{\triangle ABC}=2pR.