Question

Cho tam giác ABC có trung tuyến AD. Gọi M là trung điểm của AD , lấy N sao cho \(\overrightarrow{AC}\)= 3 \(\overrightarrow{AN}\) . Chứng minh B,N,M thẳng hàng

Answer 1

\(\overrightarrow{BM}=\frac{1}{2}\left(\overrightarrow{BA}+\overrightarrow{BD}\right)=\frac{1}{2}\overrightarrow{BA}+\frac{1}{4}\overrightarrow{BC}=\frac{1}{4}\left(2\overrightarrow{BA}+\overrightarrow{BC}\right)\left(1\right)\)

\(\overrightarrow{BN}=\overrightarrow{AN}-\overrightarrow{AB}=\frac{1}{3}\overrightarrow{AC}-\overrightarrow{AB}\)

\(=\frac{1}{3}\left(\overrightarrow{BC}-\overrightarrow{BA}\right)+\overrightarrow{BA}=\frac{1}{3}\left(2\overrightarrow{BA}+\overrightarrow{BC}\right)\left(2\right)\)

Từ \(\left(1\right);\left(2\right)\Rightarrow\text{​​}\overrightarrow{BM}=\frac{4}{3}\text{​​}\overrightarrow{BN}\) hay \(B;M;N\) thẳng hàng

Answer 2

https://i.imgur.com/H2t4tKV.png