ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that (CBSE 2004)

The figure is given below:

Given: ABCD is a trapezium

          AB ll CD

          Diagonals intersect at O

To Prove =  

Construction:  Construct a line EF through point O, such that EF is parallel to CD.

Proof: 

In ΔADC, EO is parallel to CD

According to basic propotionality theorem, if a side is drawn parallel to any side of the triangle then the corresponding sides formed are propotional

Now, using basic proportionality theorem in ΔAEO and ΔADC, we obtain

In ΔABD, OE is parallel to AB

So, using basic proportionality theorem in ΔEOD and ΔABD, we get

From (i) and (ii), we get 

Therefore by cross multiplying we get,

 

Hence, Proved