M

N

P

BC

CA

AB

ABC

AM

BN

CP

M'

N'

P'

MM'\cdot NN'\cdot PP'\leqslant\frac{R^{2}r}{4}

BC=a

\angle BAC=\alpha

\angle ABC=\beta

\angle ACB=\gamma

\angle BAM=\alpha_{1}

\angle CAM=\alpha_{2}

ABM

CBM

\frac{BM}{AM}=\frac{\sin\alpha_{1}}{\sin\beta}~\Rightarrow~BM=AM\cdot\frac{\sin\alpha_{1}}{\sin\beta},

\frac{CM}{AM}=\frac{\sin\alpha_{2}}{\sin\gamma}~\Rightarrow~CM=AM\cdot\frac{\sin\alpha_{2}}{\sin\gamma}.

BM\cdot MC=AM\cdot\frac{\sin\alpha_{1}}{\sin\beta}\cdot AM\cdot\frac{\sin\alpha_{2}}{\sin\gamma}=AM^{2}\cdot\frac{\sin\alpha_{1}\sin\alpha_{2}}{\sin\beta\sin\gamma},

AM=\sqrt{BM\cdot CM}\cdot\frac{\sin\beta\sin\gamma}{\sin\alpha_{1}\sin\alpha_{2}}.

a=2R\sin\alpha

BM\cdot MC\leqslant\frac{(BM+MC)^{2}}{4}=\frac{a^{2}}{4}=R^{2}\sin^{2}\alpha,

\sin\alpha_{1}\sin\alpha_{2}=\frac{1}{2}(\cos(\alpha_{1}-\alpha_{2})-\cos(\alpha_{1}+\alpha_{2}))\leqslant

\leqslant\frac{1}{2}(1-\cos(\alpha_{1}+\alpha_{2}))=\sin^{2}\frac{\alpha_{1}+\alpha_{2}}{2}=\sin^{2}\frac{\alpha}{2},

AM\cdot MM'=BM\cdot MC,

MM'=\frac{BM\cdot MC}{AM}=\frac{AM^{2}\sin\alpha_{1}\sin\alpha_{2}}{AM\sin\beta\sin\gamma}=\frac{AM\sin\alpha_{1}\sin\alpha_{2}}{\sin\beta\sin\gamma}=

=\sqrt{BM\cdot CM}\cdot\sqrt{\frac{\sin\beta\sin\gamma}{\sin\alpha_{1}\sin\alpha_{2}}}\cdot\frac{\sin\alpha_{1}\sin\alpha_{2}}{\sin\beta\sin\gamma}=

=\sqrt{BM\cdot CM}\cdot\sqrt{\frac{\sin\alpha_{1}\sin\alpha_{2}}{\sin\beta\sin\gamma}}\leqslant\frac{R\sin\alpha\sin\frac{\alpha}{2}}{\sqrt{\sin\beta\sin\gamma}}.

NN'\leqslant\frac{R\sin\beta\sin\frac{\beta}{2}}{\sqrt{\sin\alpha\sin\gamma}},~PP'\leqslant\frac{R\sin\gamma\sin\frac{\gamma}{2}}{\sqrt{\sin\alpha\sin\beta}}.

MM'\cdot NN'\cdot PP'\leqslant\frac{R^{3}\sin\alpha\sin\beta\sin\gamma\cdot\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\sin\frac{\gamma}{2}}{{\sqrt{\sin^{2}\alpha\sin^{2}\beta\sin^{2}\gamma}}}=

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

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