Question

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

Answer 1

Answer:

As, opposite angles are supplementary , so B , C , D and E are concyclic .

Step-by-step explanation:

To prove -  B,C,D,E are concyclic .

Proof -  In Δ ADE

              AD  =  AE

       =>  ∠ADE  = ∠AED    (angles opposite to equal sides are    equal)

       Also,  ∠ADE + ∠BDE = ∠AED + ∠DEC     (Because linear pair is 180° )

       =>  ∠BDE = ∠DEC

       So , in quadrilateral BDEC

    ∠B + ∠C + ∠BDE + ∠DEC = 360°   (As, sum of angles in quadrilateral is of 360°)

      =>  2∠B  + 2∠DEC  =  360°

      =>  ∠B + ∠DEC  =  180°

Also, according to theorem, if opposite angles are supplementary , points are concyclic .

Thus, B,C,D,E are concyclic .

Hence proved .

Answer 2

Answer:

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

In  triangle ADE, since AD = AE, we have

Thus in triangle ABC and ADE, we have,

and

Using equations (1) and (2), the above equations become

and

If the sum of any pair of opposite angles of a quadrilateral is 180 degrees, then the quadrilateral is cyclic.

Since the anglesare opposite angles of the quadrialteral BCED, then the quadrilateral is cyclic.