ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA=ED. Prove that:
(i) AD||BC (ii) EB=EC
Given that, ABCD is a cyclic quadrilateral in which
(i) Since,
EA = ED
Then,
∠EAD = ∠EDA (i) (Opposite angles to equal sides)
Since, ABCD is a cyclic quadrilateral
Then,
∠ABC + ∠ADC = 180o
But,
∠ABC + ∠EBC = 180o (Linear pair)
Then,
∠ADC = ∠EBC (ii)
Compare (i) and (ii), we get
∠EAD = ∠EBC (iii)
Since, corresponding angles are equal
Then,
BC ‖ AD
(ii) From (iii), we have
∠EAD = ∠EBC
Similarly,
∠EDA = ∠ECB (iv)
Compare equations (i), (iii) and (iv), we get
∠EBC = ∠ECB
EB = EC (Opposite angles to equal sides)