Question

In the adjoining figure. ABC is a triangle in which AB = AC. If D and E are points on AB and AC respectively such that AD = AE, show that the points B, C, E and D are concyclic.

Answer 1

Given: ABC is a triangle in which AB = AC. D and E are points on AB and AC respectively such that AD = AE

To prove:

AB = AC (given)

⇒(AB-AD) = (AC-AE)

⇒ DE = EC

⇒AD/AE=DB/E
(each equal to 1)

⇒ DE || BC (by the converse of Thales theorem)

∠DEC + ∠ECB = 180 degree

⇒∠DEC + ∠CBD =180 degree
[AB = AC ⇒ ∠C = ∠B]

Quadrilateral BCEA is cyclic

Hence, the point B, C, E, D are concyclic