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In the figure, ABCD is a trapezium in which AB∥DC. E is the midpoint of AD and F is a point on BC such that EF ∥ DC. Show that F bisects BC. ​  

join a diagonal, say AC to intersect AC at M. In triangle ADC, E is the mid point of  AD and EM is parallel to DC. By mid-point theorem we know that M is the mid-point of AC. Now in triangle ABC, M is the mid-point of AC and MF is parallel to DC. By applying mid-point theorem, we can say that F is the mid-point of BC.
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join bd ,and the point it intersects ef is o
in triangle abd
​ab| |cd(given)
so ef| |ab
e is the midpoint of ab(given)
so bo=od(converse of midpoint theorem)
in triangle bdc
of| |dc(ef| | dc)
bo=od(proved above)
so bf=fc(converse of midpoint theorem)
so f bisects bc
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