ABC

G

\angle BAC=60^{\circ}

\angle BGC=120^{\circ}

BC=a

CA=b

AB=c

GB=m

GC=n

B_{1}

AC

a^{2}=b^{2}+c^{2}-2bc\cos60^{\circ}=b^{2}+c^{2}-bc.

m^{2}=GB^{2}=\left(\frac{2}{3}BB_{1}\right)^{2}=\frac{4}{9}BB_{1}^{2}=\frac{4}{9}\cdot\frac{1}{4}(2a^{2}+2c^{2}-b^{2})=

=\frac{1}{9}(2b^{2}+2c^{2}-2bc+2c^{2}-b^{2})=\frac{1}{9}(3c^{2}+c^{2}-2bc+b^{2})=\frac{1}{9}(3c^{2}+(c-b)^{2}).

n^{2}=\frac{1}{9}(3b^{2}+(c-b)^{2}).

S_{\triangle BGC}=\frac{1}{3}S_{\triangle ABC}

\frac{1}{2}bc\sin60^{\circ}=\frac{1}{3}\cdot\frac{1}{2}mn\sin120^{\circ}~\Rightarrow~bc=3mn~\Rightarrow~b^{2}c^{2}=9m^{2}n^{2}~\Rightarrow~9b^{2}c^{2}=81m^{2}n^{2}=

=(3c^{2}+(c-b)^{2})(3b^{2}+(c-b)^{2})=9b^{2}c^{2}+(3c^{2}+3b^{2})(c-b)^{2}+(c-b)^{4}=

=9b^{2}c^{2}+(c-b)^{2}(3c^{2}+3b^{2}+c^{2}-2bc+b^{2})=9b^{2}c^{2}+(c-b)^{2}(4c^{2}-2bc+4b^{2})=

=9b^{2}c^{2}+2(c-b)^{2}(2c^{2}-bc+2b^{2})~\Rightarrow~(c-b)^{2}(2c^{2}-bc+2b^{2})=0,

2c^{2}-bc+2b^{2}\gt0

b

c

b=c

ABC