ABC

S

A_{1}

B_{1}

C_{1}

I

BC

CA

AB

A_{1}B_{1}C_{1}

S_{1}

4S_{1}=AI\cdot A_{2}B\cdot S.

A

B

C

ABC

\alpha

\beta

\gamma

r

ABC

R=1

\angle B_{1}IC_{1}-180^{\circ}-\alpha

S_{\triangle B_{1}IC_{1}}=\frac{1}{2}IB_{1}\cdot IC_{1}\sin\alpha=\frac{1}{2}r^{2}\sin\alpha.

S_{\triangle C_{1}IA_{1}}=\frac{1}{2}r^{2}\sin\beta,~S_{\triangle A_{1}IB_{1}}=\frac{1}{2}r^{2}\sin\gamma.

S_{1}=S_{\triangle A_{1}B_{1}C_{1}}=S_{\triangle B_{1}IC_{1}}+S_{\triangle C_{1}IA_{1}}+S_{\triangle A_{1}IB_{1}}=

=\frac{r^{2}}{2}(\sin\alpha+\sin\beta+\sin\gamma).

BC=2R\sin\alpha=2\sin\alpha,~CA=2\sin\beta,~AM=2\sin\gamma,

S=S_{\triangle ABC}=r\cdot\frac{BC+AC+AB}{2}=r(\sin\alpha+\sin\beta+\sin\gamma).

\frac{S_{1}}{S}=\frac{r}{2}

I

ABC

AC_{1}I

BA_{1}I

AI=\frac{IC_{1}}{\sin\frac{\alpha}{2}}=\frac{r}{\sin\frac{\alpha}{2}},~IB=\frac{IB_{1}}{\sin\frac{\beta}{2}}=\frac{r}{\sin\frac{\beta}{2}}.

A_{2}B=A_{2}I

\angle BA_{2}I=\angle BA_{2}A=\angle ACB=\gamma,

BA_{1}I

A_{2}B=A_{2}I=\frac{\frac{1}{2}IB}{\sin\frac{1}{2}\angle BA_{2}I}=\frac{IB}{2\sin\frac{\gamma}{2}}=\frac{\frac{r}{\sin\frac{\beta}{2}}}{2\sin\frac{\gamma}{2}}=

AI\cdot A_{2}B=\frac{r}{\sin\frac{\alpha}{2}}\cdot\frac{\frac{r}{\sin\frac{\beta}{2}}}{2\sin\frac{\gamma}{2}}=\frac{r^{2}}{2\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}}.

4S_{1}=AI\cdot A_{2}B\cdot S~\Leftrightarrow~AI\cdot A_{2}B=4\cdot\frac{S_{1}}{S}~\Leftrightarrow~

~\Leftrightarrow~\frac{r^{2}}{2\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}}=4\cdot\frac{r}{2}=2r,

r=4R\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}=4\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}