ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F see figure. Show that F is the mid-point of BC.

In ΔABD,
E is mid-point of AD [∵ Given]
and EG ∥ AB [∵ Given]
Hence, G is mid-point of BD [∵ Converse of Mid-point Theorem]
Similarly,
In ΔBCD,
G is mid-point of BD [∵ Prove above]
and FG ∥ DC [∵ Given]
Hence, F is mid-point of BC [∵ Converse of Mid-point Theorem]