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20/07/2021 4:20 pm
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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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20/07/2021 4:22 pm
Here, ABCE is a cyclic quadrilateral. In a cyclic quadrilateral, the sum of the opposite angles is 180°.
∠AEC + ∠CBA = 180°
As ∠AEC and ∠AED are linear pair,
∠AEC+∠AED = 180°
∠AED = ∠CBA … (1)
We know in a parallelogram; opposite angles are equal.
So, ∠ADE = ∠CBA … (2)
Now, from equations (1) and (2) we get,
∠AED = ∠ADE
Now, AD and AE are angles opposite to equal sides of a triangle,
∴ AD = AE (proved).