In triangle #ABC# we have #M# the middle of #[BC]#.How to demonstrate that #vec(AB)+vec(AC)=2vec(AM)#?

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Answer 1 · 2017-04-29T14:00:37

See proof below

We apply the sum of 2 vectors ( In France, it is called the "relation of Chasles")

#vec(AB)=vec(AC)+ vec(CB)#

Here

#LHS=vec(AB)+vec(AC)#

#=(vec(AM)+vec(MB))+(vec(AM)+vec(MC))#

As #M# is the midpoint of #BC#

#vec(BM)=vec(MC)#

#vec(BM)=-vec(MB)=-vec(MC)#

Therefore,

#LHS=vec(AM)-vec(MC)+vec(AM)+vec(MC)#

#=vec(AM)+vec(AM)#

#=2vec(AM)#

#=RHS#

#QED#