Abcd Is Parallelogram Circle Through B And C Intersects Produce If Necessary At E

ABCD is a parallelogram. The circle through A, B and C intersects (produce if necessary) at E. Prove that AE =AD.

Answers:

Given : ABCD is a parallelogram. Circle through A,B and C intersects CD produced in E. 

To Prove : AE = AD

Proof : ABCE is a cyclic quadrilateral. 

∴  ∠B + ∠E = 180°  …i)

ABCD is a parallelogram .

∴  ∠B + ∠E = ∠B + ∠2 

⇒  ∠E = ∠2  

In △ADE, we have 

 ∠E = ∠2  

⇒ AD = AE  [sides opposite to equal angles of a △]