Question

Cho \(\Delta ABC\) và \(\Delta A_1B_1C_1\)

CMR: Nếu \(\overrightarrow{AA_1}+\overrightarrow{BB_1}+\overrightarrow{CC_1}=\overrightarrow{0}\) thì hai tam giác có cùng trọng tâm

Answer 1

Gọi G là trọng tâm \(\Delta ABC\)

\(\Rightarrow\overrightarrow{AA_1}+\overrightarrow{BB_1}+\overrightarrow{CC_1}=\overrightarrow{0}\)

\(\Leftrightarrow\overrightarrow{AG}+\overrightarrow{GA_1}+\overrightarrow{BG}+\overrightarrow{GB_1}+\overrightarrow{CG}+\overrightarrow{GC_1}=\overrightarrow{0}\)

\(\Leftrightarrow\left(\overrightarrow{AG}+\overrightarrow{BG}+\overrightarrow{CG}\right)+\left(\overrightarrow{GA_1}+\overrightarrow{GB_1}+\overrightarrow{GC_1}\right)=\overrightarrow{0}\)

\(\Leftrightarrow\overrightarrow{GA_1}+\overrightarrow{GB_1}+\overrightarrow{GC_1}=\overrightarrow{0}\)

\(\Rightarrow G\) là trọng tâm \(\Delta A_1B_1C_1\) \(\left(đpcm\right)\)