Question

Chứng Minh:

 \(\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=0\)  thì \(G\) là trọng tâm \(\Delta ABC\)

Answer 1

Gọi M là trung điểm BC \(\Rightarrow\overrightarrow{MB}+\overrightarrow{MC}=\overrightarrow{0}\)

Ta có:

\(\overrightarrow{GA}+\overrightarrow{GB}+\overrightarrow{GC}=\overrightarrow{0}\)

\(\Leftrightarrow\overrightarrow{GM}+\overrightarrow{MA}+\overrightarrow{GM}+\overrightarrow{MB}+\overrightarrow{GM}+\overrightarrow{MC}=\overrightarrow{0}\)

\(\Leftrightarrow3\overrightarrow{GM}+\overrightarrow{MA}=\overrightarrow{0}\)

\(\Leftrightarrow\overrightarrow{GM}=\dfrac{1}{3}\overrightarrow{AM}\)

\(\Leftrightarrow\overrightarrow{GA}+\overrightarrow{AM}=\dfrac{1}{3}\overrightarrow{AM}\)

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

\(\Rightarrow G\) là trọng tâm tam giác ABC