I

ABC

R

R^{3}\geqslant IA\cdot IB\cdot IC

A_{1}

B_{1}

C_{1}

BC

AC

AB

ABC

AC_{1}=AB_{1}=x,~BC_{1}=BA_{1}=y,~CB_{1}=CA_{1}=z,

BC=a,~AC=b,~AB=c,~\angle BAC=\alpha,~\angle ABC=\beta,~\angle ACB=\gamma.

AB_{1}I

BA_{1}I

CB_{1}I

IA=\frac{x}{\cos\frac{\alpha}{2}},~IB=\frac{y}{\cos\frac{\beta}{2}},~IC=\frac{z}{\cos\frac{\gamma}{2}},

R=\frac{y+z}{2\sin\alpha},~R=\frac{x+z}{2\sin\beta},~R=\frac{x+y}{2\sin\gamma}.

\frac{(x+y)(y+z)(x+z)}{8\sin\alpha\sin\beta\sin\gamma}\geqslant\frac{xyz}{\cos\frac{\alpha}{2}\cos\frac{\beta}{2}\cos\frac{\gamma}{2}},

(x+y)(y+z)(x+z)\geqslant64\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}.

\sin^{2}\frac{\alpha}{2}=\frac{1-\cos\alpha}{2}=\frac{1}{2}\left(1-\frac{b^{2}+c^{2}-a^{2}}{2bc}\right)=\frac{1}{2}\cdot\frac{2bc-b^{2}-c^{2}+a^{2}}{2bc}=

=\frac{a^{2}-(b-c)^{2}}{4bc}=\frac{(a+c-b)(a+b-c)}{4bc}=\frac{(p-b)(p-c)}{bc}=\frac{yz}{(x+y)(x+z)}.

\sin^{2}\frac{\beta}{2}=\frac{xz}{(x+y)(y+z)},~\sin^{2}\frac{\gamma}{2}=\frac{xy}{(x+z)(y+z)},

\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}=\frac{xyz}{(x+y)(y+z)(x+z)}.

(x+y)(y+z)(x+z)\geqslant64\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}~\Leftrightarrow~(x+y)(y+z)(x+z)\geqslant8xyz.

x+y\geqslant2\sqrt{xy},~y+z\geqslant2\sqrt{yz},~x+z\geqslant2\sqrt{xz}.

ABC

r

IA=\frac{r}{\sin\frac{\alpha}{2}},~IB=\frac{r}{\sin\frac{\beta}{2}},~IC=\frac{r}{\sin\frac{\gamma}{2}},~R=4r\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}

IA\cdot IB\cdot IC=\frac{r^{3}}{\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}}=\frac{4r^{3}R}{r}=4r^{2}R\leqslant R^{2}\cdot R=R^{3},

2r\leqslant R