Consider a quadrilateral ABCD. K, L, M, N are the midpoints of the segments AB, BC, CD, DA respectively. O is the intersection point of LN, KM. Let P and Q be the middle points of the diagonals AC and BD respectively. Show that O is the midpoint of the PQ.
I am running into trouble with trying to describe the intersection of LN and KM using vectors. Is there an easier method that doesnt even use vectors?
A hint: The point $O$ can be written in two ways: Since $O$ is the intersection of the two lines $L\vee N$ and $K\vee M$ there are uniquely determined $s$, $t\in[0,1]$ such that $$O=(1-s){A+B\over2}+s{C+D\over2}=(1-t){B+C\over2}+t{D+A\over2}\ .$$ Here $s$ and $t$ can immediately be guessed.