Vector Proof Centroid Theorem

  • Thread starter pjallen58
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I am trying to figure out how to prove the 2:1 ratio of a triangle's medians at the centroid using vectors. Example if I had a triangle ABC with midpoints D of BC, E of AC and F of AB. I know G is where the medians intersect. I have seen many proofs and understand the process that proves the addition of the vectors from the centroid to the vertices are zero i.e. GA+GB+GC=0.
vectorsum_dreieck.GIF

Does this prove the 2:1 ratio? I cannot find anything explaining how to prove the actual 2:1 ratio. I am not sure if I am missing something or what. Any help or suggestions would be appreciated. Thanks.
 

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  • #2
tiny-tim
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I am trying to figure out how to prove the 2:1 ratio of a triangle's medians at the centroid using vectors. Example if I had a triangle ABC with midpoints D of BC, E of AC and F of AB. I know G is where the medians intersect. I have seen many proofs and understand the process that proves the addition of the vectors from the centroid to the vertices are zero i.e. GA+GB+GC=0.
View attachment 14402
Does this prove the 2:1 ratio? I cannot find anything explaining how to prove the actual 2:1 ratio. I am not sure if I am missing something or what. Any help or suggestions would be appreciated. Thanks.

Hi pjallen58! :smile:

Yes, because, for example, a + 2(d) is 1/3 of the way,

and that's a + 2(1/2(b + c)), = a + b + c! :smile:
 

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