Need some help, vector proofs

Quote:

Originally Posted by eekoz

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) ?
Thanks in advance!

Hello,

1. I've attached a diagram to demonstrate what I've calculated.

2. You ought to know that the vector $(OM_{BC})=\frac{1}{2}((OB)+(OC))$

3. You get three equations, which always describe the vector (OG):

$(OG)=(OA)+k \cdot \left(\frac{1}{2} \cdot ((OB)+(OC))-(OA) \right)$

$(OG)=(OB)+t \cdot \left(\frac{1}{2} \cdot ((OA)+(OC))-(OB) \right)$

$(OG)=(OC)+s \cdot \left(\frac{1}{2} \cdot ((OB)+(OA))-(OC) \right)$

Expand the RHS of this equations. You'll get a system of three linear equations. Solve for k, t, s. You'll get k = t = s = 2/3.

If you plug in this value into one of the equations above and you'll get your proof.

Hope that this is of some help.

Greetings

EB