Two triangles ABC and DBC lie on same side of BC such that PQ parallel BA and PR parallel BD. Prove that QR parallel AD.

Answer :
Given
$∆$ ABC and $∆$ DBC have same side BC 
And
PR | | BD 
PQ | | BA



In $∆$ ABC  PQ | | BA

So , by Basic proportionality theorem we get

$\frac{PC}{BP} = \frac{CQ}{AQ}$   ------------------- ( 1 )

In $∆$ DBC  PR | | BD

So , by Basic proportionality theorem we get

$\frac{RC}{RD} = \frac{PC}{BP}$     -------- ( 2 )

From equation 1 and 2 , we get

$\frac{RC}{RD} = \frac{CQ}{AQ}$ 

That is the ratio of side in ​In $∆$ ADC  , that is true only in one case as by Basic proportionality theorem we get

AD  | | QR                                                ( Hence proved )