A parallelogram and a line joining a vertex to the midpoint of opposite side

Question

In a parallelogram ABCD. M is the midpoint of CD. Line BM intersects AC at L and it also intersects AD extended at E. Prove that EL=2BL

PS: This is not a homework problem. I was solving geometry for fun. I'm unable to solve this. :(

Answer 1

Here is a different way to look at it.

Make two copies of the parallelogram, sharing the side CD.

The second copy is A'B'DC.

We will show that B'=E.

Consider $\triangle{B'AB}$ and $\triangle{B'DM}$. These are similar and since B'D = AD, we have that DM = AB/2. M is the midpoint of CD and thus B' = E.

Thus B'L' = BL.

Now consider $\triangle{B'DL'}$ and $\triangle{B'AL}$. These are similar and so B'L' = LL', as B'D = AD.

Thus EL = EL' + LL' = 2B'L' = 2BL.

Answer 2

Let BD and AC intersects at point O. We have triangle BCD. BO=OD because ABCD is parallelogram. In this triangle CO and BM are medians. Medians intersects in one point and divide each other in relation 2:1. So BL = 2LM.

Also BM = ME thus triangle MBC = triangle MDE because they are congruent and DM = CM.

So BL = 2/3 BM => BL = 2EL.