S

a

b

c

r

p

\mbox{а)}~3\sqrt{3}r^{2}\leqslant S\leqslant\frac{p^{2}}{3\sqrt{3}};~\mbox{б)}~S\leqslant\frac{a^{2}+b^{2}+c^{2}}{4\sqrt{3}}.

\frac{S^{2}}{p}=(p-a)(p-b)(p-c)\leqslant\left(\frac{p-a+p-b+p-c}{3}\right)^{3}=\frac{(3p-2p)^{2}}{27}=\frac{p^{3}}{27},

S^{2}\leqslant\frac{p^{4}}{27}

S\leqslant\frac{p^{2}}{3\sqrt{3}}

S=pr

pr\leqslant\frac{p^{2}}{3\sqrt{3}}

r\leqslant\frac{p}{3\sqrt{3}}

r

r^{2}\leqslant\frac{pr}{3\sqrt{3}}=\frac{S}{3\sqrt{3}}

S\geqslant3\sqrt{3}r^{2}

a

b

c

(a+b+c)^{2}\leqslant3(a^{2}+b^{2}+c^{2}),

(a+b+c)^{2}\leqslant3(a^{2}+b^{2}+c^{2})~\Leftrightarrow~2a^{2}+2b^{2}+2c^{2}-2ab-2bc-2ac\geqslant0~\Leftrightarrow

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

S\leqslant\frac{p^{2}}{3\sqrt{3}}=\frac{(a+b+c)^{2}}{4\cdot3\sqrt{3}}\leqslant\frac{3(a^{2}+b^{2}+c^{2})}{12\sqrt{3}}=\frac{a^{2}+b^{2}+c^{2}}{4\sqrt{3}}.

AB

ABC

BC=a

AC=b

AB=a

S

ABC_{1}

CC_{1}^{2}=\frac{1}{2}(a^{2}+b^{2}+c^{2})-2S\sqrt{3}

\frac{1}{2}(a^{2}+b^{2}+c^{2})-2S\sqrt{3}\geqslant0.

S\leqslant\frac{a^{2}+b^{2}+c^{2}}{4\sqrt{3}}.