AA_{1}

BB_{1}

CC_{1}

ABC

BC=a

AC=b

AB=c

a(b+c)\overrightarrow{AA_{1}}+b(a+c)\overrightarrow{BB_{1}}+c(a+b)\overrightarrow{CC_{1}}=\overrightarrow{0}.

\frac{BA_{1}}{A_{1}C}=\frac{AB}{AC}=\frac{c}{b},~BA_{1}=\frac{ac}{b+c},~A_{1}C=\frac{ab}{b+c},

AC_{1}=\frac{ac}{a+b},~BC_{1}=\frac{ac}{a+b},~AB_{1}=\frac{bc}{a+c},~CB_{1}=\frac{ab}{a+c}.

\frac{IA_{1}}{IA}=\frac{BA_{1}}{AB}=\frac{\frac{ac}{b+c}}{c}=\frac{a}{b+c},~\frac{IA_{1}}{AA_{1}}=\frac{a}{a+b+c}.

\frac{BI}{BB_{1}}=\frac{a+c}{a+b+c},~\frac{CI}{CC_{1}}=\frac{a+b}{a+b+c}.

\overrightarrow{AA_{1}}=\frac{a+b+c}{a}\overrightarrow{IA_{1}}=-\frac{a+b+c}{a}\left(\frac{b}{b+c}\overrightarrow{IB}+\frac{c}{b+c}\overrightarrow{IC}\right)=

=-\frac{a+b+c}{a(b+c)}(b\overrightarrow{IB}+c\overrightarrow{IC})=-\frac{a+b+c}{a(b+c)}\left(b\cdot\frac{a+c}{a+b+c}\overrightarrow{BB_{1}}+c\cdot\frac{a+b}{a+b+c}\overrightarrow{IC}\right)=

=-\frac{b(a+c)}{a(b+c)}\overrightarrow{BB_{1}}-\frac{c(a+b)}{a(b+c)}\overrightarrow{CC_{1}}.

a(b+c)\overrightarrow{AA_{1}}+b(a+c)\overrightarrow{BB_{1}}+c(a+b)\overrightarrow{AA_{1}}=\overrightarrow{0}.