Question

prove they are concyclic.!

Answer 1

Given that AB = AC. Therefore ABC is an isosceles triangle.

Also given that AD = AE.

We need to prove that B,C,D and E are concyclic.

Since AB = AC and AD = AE, we have BD = DE.

If a line divides any two sides of a triangle in the same ratio, then the line must be parallel tothe third side.

Thus, the line DE is parallel to the side BC.

In triangle ABC, since AB = AC, we have

∆ABC=∆ACB--------1

In  triangle ADE, since AD = AE, we have

∆ADE=∆ABD-------2

Thus in triangle ABC and ADE, we have,
∆A+∆ABC+∆ACB=180°
and
∆A+∆ADE+∆AED=180°
Using equations (1) and (2), the above equations become
∆A+2∆ACB=180°
and
∆A+2∆ADE=180°

=∆ACB=∆ADE
=∆ECB=∆AD
=ECB+EDB=ADE+EDB(adding EDB on both sides)
=ECB+EDB=180°(ADE and EDB are linear pair)
If the sum of any pair of opposite angles of a quadrilateral is 180 degrees, then the quadrilateral is cyclic.

Since the angles,
ECB and EDB
are opposite angles of the quadrialteral BCED, then the quadrilateral is cyclic.
HOPING IT WILL BE HELPFUL.....