Question

In isosceles triangle ABC,AB =AC DandEare point on BC such that BE=CD SHOW THAT AD=AE

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



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.

Answer 2

We have, AB= AC, thus angleB = angleC
In triangle, ABE and ACD
AB = AC
angle B = C
BE = CD
therefore, triangle ABE = ACD
thus, AD = AE ( CPCT )

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