Question

Cho tam giác ABC 3 trung tuyến AM BN CP và G là trọng tâm .A) chứng minh AG=1/3AB+1/3AC b)cho/_\A'B'C có trọng tâm G chứng minh rằng GG'=1/3(AA'+BB'+CC')

Answer 1

Ta có:

\(\overrightarrow{AG}=\dfrac{2}{3}\overrightarrow{AM}\)

Mà \(\overrightarrow{AM}=\dfrac{1}{2}\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{AC}\)

\(\Rightarrow\overrightarrow{AG}=\dfrac{2}{3}\left(\dfrac{1}{2}\overrightarrow{AB}+\dfrac{1}{2}\overrightarrow{AC}\right)=\dfrac{1}{3}\overrightarrow{AB}+\dfrac{1}{3}\overrightarrow{AC}\)

\(\dfrac{1}{3}\left(\overrightarrow{AA'}+\overrightarrow{BB'}+\overrightarrow{CC'}\right)=\dfrac{1}{3}\left(\overrightarrow{AG}+\overrightarrow{GG'}+\overrightarrow{G'A'}+\overrightarrow{BG}+\overrightarrow{GG'}+\overrightarrow{G'B'}+\overrightarrow{CG}+\overrightarrow{GG'}+\overrightarrow{G'C'}\right)\)

\(=\dfrac{1}{3}.3.\overrightarrow{GG'}=\overrightarrow{GG'}\)