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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.
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