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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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Data : ABCD is a parallelogram. 

The circle through A, B and C intersect CD at E. AE is joined. 

To Prove: AE = AD 

Proof: ∠AEC + ∠AED = 180° …………. (i) (linear pair) 

ABCE is a cyclic quadrilateral. 

∴ ∠ABC + ∠AEC = 180° ………….(ii) 

(opposite angles) Comparing (i) and (ii), 

∠AEC = ∠AED = ∠ABC + ∠AEC 

∠AED = ∠ABC ………….. (iii) 

But, ∠ABC = ∠ADE (Opposite angles of quadrilateral) 

Substituting in equation (iii), 

∠AED = ∠ADE 

∴ AE = AD.

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