Question

Cho hai tam giác \(ABC\) và \(A_1B_1C_1\) có cùng trọng tâm \(G\). Gọi \(G_1;G_2;G_3\) lần lượt là trọng tâm tam giác \(BCA_1;ABC_1;ACB_1\) . Chứng minh rằng  \(\overrightarrow{GG_1}+\overrightarrow{GG_2}+\overrightarrow{GG_3}=\overrightarrow{0}\)

Answer 1

Từ giả thiết \(G_1\) là trọng tâm \(BCA_1\)

\(\overrightarrow{G_1B}+\overrightarrow{G_1C}+\overrightarrow{G_1A_1}=\overrightarrow{0}\)

\(\Rightarrow\overrightarrow{G_1G}+\overrightarrow{GB}+\overrightarrow{G_1G}+\overrightarrow{GC}+\overrightarrow{G_1G}+\overrightarrow{GA_1}=0\)

\(\Leftrightarrow\overrightarrow{GB}+\overrightarrow{GC}+\overrightarrow{GA_1}=3\overrightarrow{GG_1}\)

Tương tự:

\(\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC_1}=3\overrightarrow{GG_2}\)

\(\overrightarrow{GA}+\overrightarrow{GC}+\overrightarrow{GB_1}=3\overrightarrow{GG_3}\)

Cộng vế:

\(3\left(\overrightarrow{GG_1}+\overrightarrow{GG_2}+\overrightarrow{GG_3}\right)=2\left(\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}\right)+\left(\overrightarrow{GA_1}+\overrightarrow{GA_2}+\overrightarrow{GA_3}\right)=2.\overrightarrow{0}+\overrightarrow{0}=\overrightarrow{0}\)

\(\Rightarrow\overrightarrow{GG_1}+\overrightarrow{GG_2}+\overrightarrow{GG_3}=\overrightarrow{0}\)