ABCD is a parallelogram in which E is the midpoint of AD. DL || EB meeting AB produced at F and BC at L. Prove that DF = 2 DL

To Prove: DF = 2DL

Given: ABCD is a parallelogram. E is the midpoint of AD. DL || EB meeting AB produced at F and BC at L.

Concept Used:

Converse Midpoint Theorem: A line drawn through the mid-point of any side of a triangle and parallel to another side, bisects the third side.

Opposite sides of a parallelogram are parallel and equal to each other.

Diagram:

Proof:

In ΔADF,

E is the midpoint of AD and EB || DF

B is the mid-point of AF [Converse mid-point theorem]

In ΔADF,

E and B are the midpoints AD and AF respectively

By mid-point theorem, we have,

EB = 1/2 DF.........(1)

Since EB || DL and BL || ED.

Opposite sides are parallel, and therefore, EBLD is a parallelogram.

EB = DL......(2) [opposite sides of a parallelogram are equal]

DL = 1/2 DF [L is the midpoint of DF]

LF = DL = EB

Therefore, EB = LF

Now, 1/2 DF = DL.

Hence, DF = 2 DL

Hence, Proved.