September 25th, 2013, 01:25 AM  #1 
Member Joined: Jul 2009 Posts: 34 Thanks: 0  2D vector proofing
A, B and C are the midpoints of PQ, QR, RQ respectively. G is the point of intersection of AR, BP and CQ. Given that the position vectors of P, Q and R relative to an origin O, (which is not shown in the diagram), are p, q and r respectively, prove that p + q + r = 3g, where g is the position vector of G relative to the origin O.

September 25th, 2013, 02:51 PM  #2 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: 2D vector proofing
You will have to keep the sketch in your view while examining this edited proof. Comment if any statement I made needs better explanation. Proof starts here Claim: Proof of claim By completing the parallelogram in the image with base and slant height , we observe that . But also observe that . This is because each of the segments divide the triangle into two halves equal in area. It is then easy to deduce that each of the six triangular segments in the diagram have the same area; and then that the area of is onethird that of . Hence, by (1) But also, By validity of (2) and (3) we get that , concluding the proof of our claim. The following relations hold: Adding them up, we get that 

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