Question

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.

Answer 1

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]

In ∆ABC,

∴ `(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.