D and E are points on equal sides AB and AC of an isosceles triangle ABC such that AD = AE. Prove that the points B, C, E and D are concyclic.
Given – In ∆ABC, AB = AC and D and E are points on AB and AC
Such that AD = AE. DE is joined.
To prove B, C, E, D are concyclic.
Proof – In ∆ABC, AB = AC
∴ ∠B = ∠C [Angles opposite to equal sides]
Similarly, In ∆ADE, AD = AE [Given]
∴ ∠ADE = ∠AED [Angles opposite to equal sides]
∴ `(AD)/(AB) = (AE )/(AC)`
DE || BC
∴ ∠ADE = ∠B [corresponding angles]
But ∠B = ∠C [proved]
∴ Ext ∠ADE = its interior opposite ∠C
∴ BCED is a cyclic quadrilateral
Hence B, C, E and D are concyclic.