Bonjour,
1)
[tex]D=mil[BC] \Longrightarrow \overrightarrow{BD}=\overrightarrow{DC}\\\\\overrightarrow{AB}.\overrightarrow{AC}=(\overrightarrow{AD}+\overrightarrow{DB}).(\overrightarrow{AD}+\overrightarrow{DC})\\\\=AD^2+\overrightarrow{DB}.\overrightarrow{AD}+\overrightarrow{AD}.\overrightarrow{DC}-DC^2\\\\=AD^2-DC^2\\[/tex]
2)
[tex]\overrightarrow{AB}.\overrightarrow{AC}=AD^2-DC^2\\\\=AB^2-BD^2-DC^2(Pythagore)\\\\=AB^2-2DC^2\\[/tex]
3)
[tex]\overrightarrow{AB}.\overrightarrow{AC}=3^2-2*2^2=1=3*3*cos(\widehat{BAC})\\\\cos(\widehat{BAC})=\dfrac{1}{9} \Longrightarrow\ \widehat{BAC}=83.620629...^ o\approx84^o\\[/tex]
