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

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 Given : ABCD is a parallelogram. 

To Prove : AE = AD. 

Construction : Draw a circle which passes through ABC and intersect CD (or CD produced) at E. 

Proof : For fig (i) 

∠AED + ∠ABC = 180° [Linear pair] ... (ii) 

But ∠ACD = ∠ADC = ∠ABC + ∠ADE 

⇒ ∠ABC + ∠ADE = 180° [From (ii)] ... (iii) 

From (i) and (iii) 

∠AED + ∠ABC = ∠ABC + ∠ADE 

⇒ ∠AED = ∠ADE 

⇒ ∠AD = ∠AE [Sides opposite to equal angles are equal] 

Similarly we can prove for Fig (ii) Proved.

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