a: \(\overrightarrow{BI}=\overrightarrow{BA}+\overrightarrow{AI}\)
\(=\overrightarrow{BA}+\frac12\cdot\overrightarrow{AM}=\overrightarrow{BA}+\frac12\cdot\frac12\cdot\left(\overrightarrow{AB}+\overrightarrow{AC}\right)\)
\(=-\overrightarrow{AB}+\frac14\cdot\overrightarrow{AB}+\frac14\cdot\overrightarrow{AC}=-\frac34\cdot\overrightarrow{AB}+\frac14\cdot\overrightarrow{AC}\)
\(=-\frac14\cdot\left(3\cdot\overrightarrow{AB}-\overrightarrow{AC}\right)\)
\(\overrightarrow{BK}=\overrightarrow{BA}+\overrightarrow{AK}=\overrightarrow{BA}+\frac13\cdot\overrightarrow{AC}\)
\(=\frac13\left(3\cdot\overrightarrow{BA}+\overrightarrow{AC}\right)=-\frac13\left(3\cdot\overrightarrow{AB}-\overrightarrow{AC}\right)\)
\(=-\frac14\cdot\frac43\left(3\cdot\overrightarrow{AB}-\overrightarrow{AC}\right)=\frac43\cdot\overrightarrow{BI}\)
=>B,I,K thăng hàng
b: Vì \(\overrightarrow{BK}=\frac43\cdot\overrightarrow{BI}\)
nên \(\frac{BI}{BK}=1:\frac43=\frac34\)
