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Vector proof question

  1. Apr 20, 2006 #1
    Okay, so I have a triagle with vertices A, B, and C.
    I know that the centroid, G, is where all the medians of the triangle intersect, and G divides the median at a 2:1 ratio

    Assuming point O is a point that's not on the triangle, how can I prove:
    OG = 1/3(OA + OB + OC) ?

    I've seen this equation a lot of times, but I'd like to see a proof of it for interest sake

    Thanks
     
  2. jcsd
  3. Apr 21, 2006 #2

    AKG

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    Let [itex]P[/itex] be the midpoint of [itex]\overline{BC}[/tex]. Then:

    [tex]\overline{OB} + \frac{1}{2}\overline{BC} = \overline{OP}[/tex]

    [tex]\overline{OB} + \overline{BC} = \overline{OC}[/tex]

    [tex]\overline{OP} + \overline{PA} = \overline{OA}[/tex]

    [tex]\overline{OP} + \frac{1}{3}\overline{PA} = \overline{OG}[/tex]

    The last line comes from the fact that G divides the median [itex]\overline{PA}[/itex] at a 2:1 ratio. You should be able to figure it out from here.
     
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