$OABC$ is a parallelogram with $O$ at the origin and $a,b,c$ are the position vectors of the points $A,B, and$ $C$.

Question

$OABC$ is a parallelogram with $O$ at the origin and $a,b,c$ are the position vectors of the points $A,B, and$ $C$. $P$ is the midpoint of $BC$ and $Q$ is the point on $OB$ such that $OQ:QB$ is 2:1. Prove $APQ$ is a straight line

I keep seeming to come round in circles in trying to prove this and it is just seeming unnecessarily messy. Any help would be appreciated.

Answer 1

If you mark midpoint of $AC$ with $R$ you see that $R$ is also a midpoint for $OB$ since $OABC$ is a paralellogram.

So $Q$ is a gravity center for $ABC$ since $BQ:QR=2:1$, so $Q$ is also on $AP$.