#### 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.

#### Solution

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.

Is there an error in this question or solution?

Solution 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. Concept: Cyclic Properties.