# Vectors, just show that two points are equal? How many approaches are there?

This is a fairly simple thing to do, but what would be the optimal approach to solve part (c) (parts (a) and (b) are done):

In the triangle ABC M is the midpoint on AB. Let OA = $\vec{a}$, OC = $\vec{c}$ and OB = $\vec{b}$.

a) Find the vector OM expressed with $\vec{a}$ and $\vec{b}$.
b) The point P is on CM so that CP = 2PM. Find OP expressed with $\vec{a}, \vec{b}$ and $\vec{c}$.
c) Let N be the midpoint on AC. The point Q is suppose to be on BN so that BQ = 2QN. Show that P = Q.

Logically, how does one go about doing this?

Would you assume that these points are not equal and try to derive a contradiction? Or would you make that point waypoint for another vector sum and show somehow that by implication they are indeed equal?

I know they are equal. In the head of the task the following definition is given; OA = $\vec{a}$, OC = $\vec{c}$ and OB = $\vec{b}$.

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The problem has been hard to understand! It boils down to a standard result about the medians of a triangle $ABC$. They all meet at a point $P$, and any median is divided by $P$ in the ratio $2:1$. The result goes back to Euclid, and probably earlier. There are nice basic geometry proofs. A vector proof is a frequent exercise in a first linear algebra course. – André Nicolas Feb 2 '12 at 21:53

First do parts (a) and (b) of the exercise. Then do them again with points $B$ and $C$ swapped, which gives you an expression for $OQ$. You will find that the expressions for $OP$ and $OQ$ are identical; hence $P$ and $Q$ must be the same point.