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a: \(\overrightarrow{BM}=\overrightarrow{BA}+\overrightarrow{AM}=\overrightarrow{BA}+\frac12\cdot\overrightarrow{AD}\)
\(=\overrightarrow{BA}+\frac12\cdot\frac12\cdot\left(\overrightarrow{AB}+\overrightarrow{AC}\right)=-\frac34\cdot\overrightarrow{AB}+\frac14\cdot\overrightarrow{AC}\)
\(=-\frac14\left(3\cdot\overrightarrow{AB}-\overrightarrow{AC}\right)\)
\(\overrightarrow{BN}=\overrightarrow{BA}+\overrightarrow{AN}=\overrightarrow{BA}+\frac13\cdot\overrightarrow{AC}=\frac{-1}{3}\left(3\cdot\overrightarrow{AB}-\overrightarrow{AC}\right)\)
=>\(\frac{\overrightarrow{BM}}{\overrightarrow{BN}}=\frac{-1}{4}:\frac{-1}{3}=\frac34\)
=>B.M,N thẳng hàng
b: \(\overrightarrow{IM}=\overrightarrow{IA}+\overrightarrow{AM}\)
\(=-\frac23\cdot\overrightarrow{AB}+\frac12\cdot\overrightarrow{AD}=-\frac23\cdot\overrightarrow{AB}+\frac12\cdot\frac12\left(\overrightarrow{AB}+\overrightarrow{AC}\right)\)
\(=-\frac23\cdot\overrightarrow{AB}+\frac14\cdot\overrightarrow{AB}+\frac14\cdot\overrightarrow{AC}=\frac{-5}{12}\cdot\overrightarrow{AB}+\frac14\cdot\overrightarrow{AC}\)
\(=\frac14\left(-\frac53\cdot\overrightarrow{AB}+\overrightarrow{AC}\right)\)
\(\overrightarrow{IJ}=\overrightarrow{IA}+\overrightarrow{AJ}=-\frac23\cdot\overrightarrow{AB}+\frac25\cdot\overrightarrow{AC}=-2\left(\frac13\cdot\overrightarrow{AB}-\frac15\cdot\overrightarrow{AC}\right)\)
\(=\frac25\left(\frac{-5}{3}\cdot\overrightarrow{AB}+\overrightarrow{AC}\right)\)
Do đó: \(\frac{\overrightarrow{IM}}{\overrightarrow{IJ}}=\frac14:\frac25=\frac58\)
=>I,M,J thẳng hàng
\(\overrightarrow{BN}=\overrightarrow{BA}+\overrightarrow{AN}=-\overrightarrow{AB}+\frac{1}{3}\overrightarrow{AC}\)
\(\overrightarrow{BM}=\overrightarrow{BA}+\overrightarrow{AM}=-\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AD}=-\overrightarrow{AB}+\frac{1}{2}\left(\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AC}\right)\)
\(\overrightarrow{BM}=-\frac{3}{4}\overrightarrow{AB}+\frac{1}{4}\overrightarrow{AC}=\frac{3}{4}\left(-\overrightarrow{AB}+\frac{1}{3}\overrightarrow{AC}\right)=\frac{3}{4}\overrightarrow{BN}\)
\(\Rightarrow B;M;N\) thẳng hàng