Question

cho tam giác ABC , M thuộc cạnh AC sao cho \(\overrightarrow{MA}=2\overrightarrow{MC}\) , \(N\in BM\) sao cho \(\overrightarrow{NB}=-3\overrightarrow{NM}\) , \(P\in BC\) sao cho \(\overrightarrow{PB}=k.\overrightarrow{PC}\) . Tìm K để 3 điểm A,N,P thẳng hàng

Answer 1

\(\overrightarrow{NB}=-3\overrightarrow{NM}\Rightarrow\frac{\overrightarrow{NB}}{\overrightarrow{NM}}=-3\)

\(\overrightarrow{MA}=2\overrightarrow{MC}\Rightarrow\overrightarrow{MA}=-2\overrightarrow{AC}\Rightarrow\frac{\overrightarrow{MA}}{\overrightarrow{AC}}=-2\)

Áp dụng định lý Menelaus cho tam giác BCM:

\(\frac{\overrightarrow{NB}}{\overrightarrow{NM}}.\frac{\overrightarrow{MA}}{\overrightarrow{AC}}.\frac{\overrightarrow{CP}}{\overrightarrow{PB}}=1\Leftrightarrow\left(-3\right).\left(-2\right).\frac{\overrightarrow{CP}}{\overrightarrow{PB}}=1\)

\(\Leftrightarrow\overrightarrow{PB}=6\overrightarrow{CP}\Rightarrow\overrightarrow{PB}=-6\overrightarrow{PC}\Rightarrow k=-6\)