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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. - ICSE Class 10 - Mathematics

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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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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.
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