SOLUTION: In trapezoid ABCD, diagonals AC,BD intersect at M. AD is parallel to BC and MN. AD=3, BC=6, determine MN. Picture Link= https://imgur.com/a/4xojP

1.  Consider triangles BMC and AMD.
    They are similar, since their interior angles are congruent in pairs
        (it is easy consequence of parallelism of lines AD and BC).

    The similarity coefficient is  = 2 from larger to smaller.


2.  It means that AM is half of the MC length. 
    It implies that AM is one third of the length AC, and that MC is two third of the length of AC.


3.  The triangles ACD and MCN are similar, too,
    and the similarity coefficient is   (from larger to smaller).

    So,   = ,

    and then, substituting |AD| = 3, you get   = ,

    which implies  |MN| = 2.