Given :
ABCD is a trapezium, where AB || DC, E is the mid-point of AD and EF || AB.
By converse of mid-point theorem,
A line drawn through the mid-point of any side of a triangle and parallel to another side bisects the third side.
Let EF intersect BD at point G.
In trapezium ABCD,
EF || AB and E is the mid-point of AD.
By converse of mid-point theorem,
G is the mid-point of DB
As, EF || AB and AB || CD,
EF || CD (Two lines parallel to the same line are parallel to each other)
In Δ BCD,
GF || CD and G is the mid-point of line BD.
By using the converse of mid-point theorem, F is the mid-point of BC.
Hence, proved that F is the mid-point of BC.