D and E are points on equal sides AB and AC of an isosceles triangle ABC such that AD=AE.Prove that B,C,D,E are cocyclic

In ΔABC,

∠ABC = ∠ACB             (Angles opposite to equal side are equal)

⇒ ∠DBC = ∠ ECB    .....(1)

 Now,

AB = AC and AD = AE

⇒ AB – AD = AC – AE

⇒ DB = EC

 ⇒ DE || BC           (B.P.T.)

 Now DE || BC and BD is the transversal

⇒ ∠DBC + ∠BDE = 180°          ....(2)        (Sum of adjacent angles is supplementary)

 From (1) and (2), we get

∠ECB + ∠BDE = 180°

⇒ BCED are concylic           (Opposide angles of a cyclic quadrilateral are supplementary)