Belorusija 2012

Two distinct points $A$ and $B$ are marked on the left half of the parabola $y=x^2$. Consider any pair of parallel lines which pass through $A$ and $B$ and intersect the right half of the parabola at points $C$ and $D$. Let $K$ be the intersection point of the diagonals $AC$ and $BD$ of the obtained trapezoid $ABCD$. Let $M,N$ be the midpoints of the bases of $ABCD$. Prove that the difference $KM-KN$ depends only on the choice of points $A$ and $B$ but does not depend on the pair of parallel lines described above. (I. Voronovich)