ABC

BC=a

CA=b

AB=c

m_{a}

m_{b}

m_{c}

G

AG

BG

CG

ABC

A_{1}

B_{1}

C_{1}

\frac{S_{\triangle A_{1}B_{1}C_{1}}}{S_{\triangle ABC}}=\frac{(a^{2}+b^{2}+c^{2})^{3}}{(8m_{a}m_{b}m_{c})^{2}}.

O

R

ABC

B_{1}C_{1}=a_{1}

C_{1}A_{1}=b_{1}

A_{1}B_{1}=c_{1}

GAB

GB_{1}A_{1}

\frac{A_{1}B_{1}}{BA}=\frac{GB_{1}}{GA},~\mbox{или}~\frac{c_{1}}{c}=\frac{GB_{1}}{GA},

GB_{1}\cdot GB=R^{2}-OG^{2}

c_{1}=c\cdot\frac{GB_{1}}{GA}=c\cdot\frac{GB_{1}\cdot GB}{GA\cdot GB}=c\cdot\frac{R^{2}-OG^{2}}{GA\cdot GB}.

a_{1}=a\cdot\frac{R^{2}-OG^{2}}{GB\cdot GC},~b_{1}=b\cdot\frac{R^{2}-OG^{2}}{GC\cdot GA}.

\frac{S_{\triangle A_{1}B_{1}C_{1}}}{S_{\triangle ABC}}=\frac{(R^{2}-OG^{2})^{3}}{(GA\cdot GB\cdot GC)^{2}}.

OG^{2}=R^{2}-\frac{1}{9}(a^{2}+b^{2}+c^{2}),

R^{2}-OG^{2}=R^{2}-\left(R^{2}-\frac{1}{9}(a^{2}+b^{2}+c^{2})\right)=\frac{1}{9}(a^{2}+b^{2}+c^{2}),

GA\cdot GB\cdot GC=\frac{2}{3}m_{a}\cdot\frac{2}{3}m_{b}\cdot\frac{2}{3}m_{c}^{2}.

\frac{S_{\triangle A_{1}B_{1}C_{1}}}{S_{\triangle ABC}}=\frac{\left(\frac{1}{9}(a^{2}+b^{2}+c^{2})\right)^{3}}{\left(\frac{2}{3}m_{a}\cdot\frac{2}{3}m_{b}\cdot\frac{2}{3}m_{c}^{2}\right)^{2}}=\frac{\frac{1}{9^{3}}(a^{2}+b^{2}+c^{2})^{3}}{\frac{1}{9^{3}}(8m_{a}m_{b}m_{c})^{2}}=\frac{(a^{2}+b^{2}+c^{2})^{3}}{(8m_{a}m_{b}m_{c})^{2}}.

\frac{a^{2}+b^{2}+c^{2}}{m_{a}^{2}+m_{b}^{2}+m_{c}^{2}}=\frac{3}{4}

\frac{S_{\triangle A_{1}B_{1}C_{1}}}{S_{\triangle ABC}}=\frac{(m_{a}^{2}+m_{b}^{2}+m_{c}^{2})^{3}}{27(m_{a}m_{b}m_{c})^{2}}.