Lời giải:
Ta có:
\(\left\{\begin{matrix} \overrightarrow{BI}=\overrightarrow{BA}+\overrightarrow{AI}\\ \overrightarrow{BI}=\overrightarrow{BM}+\overrightarrow{MI}\end{matrix}\right.\)
\(\Rightarrow 2\overrightarrow{BI}=\overrightarrow{BA}+\overrightarrow{BM}+(\overrightarrow{AI}+\overrightarrow{MI})=\overrightarrow{BA}+\overrightarrow{BM}\)
\(=\overrightarrow{BA}+\frac{\overrightarrow{BC}}{2}\)
\(\Rightarrow 4\overrightarrow{BI}=2\overrightarrow{BA}+\overrightarrow{BC}\)
Lại có:
\(\overrightarrow{BK}=\overrightarrow{BA}+\overrightarrow{AK}=\overrightarrow{BA}+\frac{\overrightarrow{AC}}{3}\)
\(\Rightarrow 3\overrightarrow{BK}=3\overrightarrow{BA}+\overrightarrow{AC}=2\overrightarrow{BA}+(\overrightarrow{BA}+\overrightarrow{AC})=2\overrightarrow{BA}+\overrightarrow{BC}\)
Do đó:
\(4\overrightarrow{BI}=3\overrightarrow{BK}\Rightarrow B,I,K\) thẳng hàng.
