Vì $N$ thuộc cạnh $BC$ thỏa $BN = 2NC$
`=>\vec{BN}=2\vec{NC}`
`<=>\vec{BN}=2.(\vec{NB}+\vec{BC})`
`<=>\vec{BN}=2\vec{NB}+2\vec{BC}`
`<=>3\vec{BN}=2\vec{BC}`
`<=>\vec{BN}=2/ 3 \vec{BC}`
+) `\vec{AN}=\vec{AB}+\vec{BN}`
`\qquad =\vec{AB}+2/ 3 \vec{BC}`
`\qquad =\vec{AB}+2/ 3 (\vec{BA}+\vec{AC})`
`\qquad =\vec{AB}- 2/ 3 \vec{AB}+2/ 3 \vec{AC}`
`\qquad=1/ 3 \vec{AB} +2/ 3 \vec{AC}`
Vậy: `\vec{AN}=1/ 3 \vec{AB} +2/ 3 \vec{AC}` (đpcm)