A_{1}

B_{1}

C_{1}

BC

AC

AB

ABC

AA_{1}

BB_{1}

CC_{1}

A_{1}C=x

B_{1}A=y

C_{1}B=z

R

ABC

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

\frac{xyz}{2R}

BC=a

AC=b

AB=c

\frac{BA_{1}}{A_{1}C}\cdot\frac{CB_{1}}{B_{1}A}\cdot\frac{AC_{1}}{C_{1}B}=1,~\mbox{или}~\frac{a-x}{x}\cdot\frac{b-y}{y}\cdot\frac{c-z}{z}=1,

xyz=(a-x)(b-y)(c-z)~\Leftrightarrow

\Leftrightarrow~xyz=abc-bcx-acy-abz+ayz+bxz+cxy-xyz~\Leftrightarrow

\Leftrightarrow~abc-bcx-acy-abz+ayz+bxz+cxy=2xyz.

ABC

S

S_{\triangle A_{1}B_{1}C_{1}}=S-(S_{\triangle AB_{1}C_{1}}+S_{\triangle BA_{1}C_{1}}+S_{\triangle CA_{1}B_{1}})=

=S-\left(\frac{AC_{1}}{AB}\cdot\frac{AB_{1}}{AC}\cdot S+\frac{BA_{1}}{BC}\cdot\frac{BC_{1}}{AB}\cdot S+\frac{CB_{1}}{AC}\cdot\frac{CA_{1}}{BC}\cdot S\right)=

=S-\left(\frac{c-z}{c}\cdot\frac{y}{b}\cdot S+\frac{b-y}{b}\cdot\frac{x}{a}\cdot S+\frac{a-x}{a}\cdot\frac{z}{c}\cdot S\right)=

=S\left(1-\frac{(c-z)y}{bc}-\frac{(b-y)x}{ab}-\frac{(a-x)z}{ac}\right)=

=\frac{S(abc-ay(c-z)-cx(b-y)-bz(a-x))}{abc}=

=\frac{S(abc-acy+ayz-bcx+cxy-abz+bzx)}{abc}=

=\frac{S(abc-bcx-acy-abz+ayz+bzx+cxy)}{abc}=\frac{2Sxyz}{abc}=\frac{xyz}{2R}

S=\frac{abc}{4R}