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

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. :(

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Can someone think of a better title? –  Aryabhata Nov 10 '10 at 22:56
No! but if you can please comment. –  claws Nov 11 '10 at 7:51
I couldn't either! Hence the request. Maybe something like "Line joining midpoint of other side". EDIT: changed. –  Aryabhata Nov 11 '10 at 15:12

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.

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