D and E are points on equal sides AB and AC of an isoscles triangle ABC such that AD - AE. Prove that the points B, C, E, D are concyclic.

Given : ABC is an isosceles triangle with AB = AC and D and E are points of side AB and AC such that AD = AE.

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°

⇒ B,C,E, D are concylic           (Opposite angles of a cyclic quadrilateral are supplementary)