AL→=AB→+mm+nBC→\overrightarrow{AL} = \overrightarrow{AB} + \frac{m}{m+n}\overrightarrow{BC}AL=AB+m+nmBC BM→=BC→+mm+nCA→=BC→+mm+n(−BC→−AB→)=−mm+nAB→+nm+nBC→\overrightarrow{BM} = \overrightarrow{BC} + \frac{m}{m+n}\overrightarrow{CA} = \overrightarrow{BC} + \frac{m}{m+n}\left(-\overrightarrow{BC} - \overrightarrow{AB}\right) = -\frac{m}{m+n}\overrightarrow{AB} + \frac{n}{m+n}\overrightarrow{BC}BM=BC+m+nmCA=BC+m+nm(−BC−AB)=−m+nmAB+m+nnBC CN→=−BC→−nm+nAB→\overrightarrow{CN} = -\overrightarrow{BC} - \frac{n}{m+n}\overrightarrow{AB}CN=−BC−m+nnAB ∴AL→+BM→+CN→=AB→+mm+nBC→−mm+nAB→+nm+nBC→−BC→−nm+nAB→=0⃗\therefore \overrightarrow{AL} + \overrightarrow{BM} + \overrightarrow{CN} = \overrightarrow{AB} + \frac{m}{m+n}\overrightarrow{BC} - \frac{m}{m+n}\overrightarrow{AB} + \frac{n}{m+n}\overrightarrow{BC} - \overrightarrow{BC} - \frac{n}{m+n}\overrightarrow{AB} = \vec{0}∴AL+BM+CN=AB+m+nmBC−m+nmAB+m+nnBC−BC−m+nnAB=0