Question

In Fig. ABCD is a cyclic quadrilateral. A circle passing through A and B meets AD and BC in the points E and F respectively. Prove that EF || DC.

Answer 1

In order to prove that EF || DC. It is sufficient to show that ∠2 = ∠3.
Since ABCD is a cyclic quadrilateral.
∴ ∠1 + ∠3 = 180°               ...(i)

Similarly, in the cyclic quadrilateral ABFE, we have
∠1 + ∠2 = 180°                  ...(ii)
 From (i) and (ii), we get 
⇒ ∠1 + ∠3 = ∠1 + ∠2
⇒ ∠3 = ∠2
Hence, EF || DC.