Question

Cho ΔABC có trọng tâm G, H là điểm đối xứng với B qua G, M là trung điểm BC.

C/M 1)  \(\overrightarrow{AH}\) = \(\dfrac{2}{3}\overrightarrow{AC}-\dfrac{1}{3}\overrightarrow{AB}\)

2) \(\overrightarrow{CH}=-\dfrac{1}{3}\overrightarrow{AB}-\dfrac{1}{3}\overrightarrow{AC}\)

3) \(\overrightarrow{MH}=\dfrac{1}{6}\overrightarrow{AC}-\dfrac{5}{6}\overrightarrow{AB}\)

 

Answer 1

H đối xứng B qua G \(\Rightarrow\overrightarrow{BH}=2\overrightarrow{BG}=2\left(\dfrac{1}{3}\overrightarrow{BA}+\dfrac{1}{3}\overrightarrow{BC}\right)=-\dfrac{2}{3}\overrightarrow{AB}+\dfrac{2}{3}\overrightarrow{BC}\)

\(\overrightarrow{AH}=\overrightarrow{AB}+\overrightarrow{BH}=\overrightarrow{AB}-\dfrac{2}{3}\overrightarrow{AB}+\dfrac{2}{3}\overrightarrow{BC}=\dfrac{1}{3}\overrightarrow{AB}+\dfrac{2}{3}\left(\overrightarrow{BA}+\overrightarrow{AC}\right)\)

\(=\dfrac{1}{3}\overrightarrow{AB}-\dfrac{2}{3}\overrightarrow{AB}+\dfrac{2}{3}\overrightarrow{AC}=\dfrac{2}{3}\overrightarrow{AC}-\dfrac{1}{3}\overrightarrow{AB}\)

\(\overrightarrow{CH}=\overrightarrow{CA}+\overrightarrow{AH}=-\overrightarrow{AC}+\dfrac{2}{3}\overrightarrow{AC}-\dfrac{1}{3}\overrightarrow{AB}=-\dfrac{1}{3}\overrightarrow{AB}-\dfrac{1}{3}\overrightarrow{AC}\)

\(\overrightarrow{MH}=\overrightarrow{MA}+\overrightarrow{AH}=-\dfrac{1}{2}\overrightarrow{AB}-\dfrac{1}{2}\overrightarrow{AC}+\dfrac{2}{3}\overrightarrow{AC}-\dfrac{1}{3}\overrightarrow{AB}\)

\(=-\dfrac{5}{6}\overrightarrow{AB}+\dfrac{1}{6}\overrightarrow{AC}\)