Given points P,Q, how do I prove (P+Q)/2 lies on the line PQ using vector geometry

Question

I had this slide in my lecture note which claims $\frac{(p+n)}{2}$ is the center point of $w=p-n$. Can someone give me a proof

Answer 1

Let $R = (P+Q)/2$ then $$PR = R-P = {Q-P\over 2}$$ and $$RQ = Q-R ={Q-P\over 2}$$ so $RQ = PR$ so $RQ$ and $PR$ are parallel so $R\in PQ$.

Answer 2

Hint: $\left(p - \cfrac{p+n}2\right) + \left(\cfrac{p+n}2 - n\right) = p - n$