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.

Question

Philoid · Accepted Answer

Given: AD = AE …(i)

& AB = AC …(ii)

Subtracting AD from both sides of equation (ii), we get

AB – AD = AC – AD

⇒ AB – AD = AC – AE [from equation (i)]

⇒ DB = EC [∵ AB – AD = DB & AC – AE = EC] …(iii)

Now, divide equation (i) by (iii), we get

By converse of Thale’s theorem, we can conclude by this equation that DE ∥ BC.

So, ∠DEC + ∠ECB = 180° [∵ sum of interior angles on the same transversal line is 180°]

Or ∠DEC + ∠DBC = 180° [∵ AB = AC ⇒ ∠C = ∠B]

Hence, we can write DEBC is cyclic and points D, E, B and C are concyclic.