L and M are the midpoints of the diagonals BD and AC respectively of the quadrilateral ABCD. Through D, draw DE equal and parallel to AB. Show that EC is parallel to LM and is double of it

Given that ABCD is a quadrilateral where L and M are mid points of AC and BD respectively.

DE is drawn parallel to AB and DE = AB
Therefore, ABED is a parallelogram.
And in parallelogram, diagonals bisect each other.
Thus AE passes through the mid point of BD, therefore AE passes through L.
So, diagonals AE and BD bisect each other.
Therefore, L is midpoint of AE and it is given that M is the mid point of AC.
Join EC
Considering the ∆ACE
L and M are the mid pints of AE and AC respectively
By mid point theorem
LM ∥ EC and LM = $\frac{1}{2}\mathrm{EC}$
Therefore, EC = 2LM

Hence proved.