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# ABCD is a parallelogram. The circle through A, B and C intersects CD produced at E, prove that AE=AD

Asked by 23rd February 2013, 9:43 PM
Answer : Given :ABCD is a parallelogram. The circle through A, B and C intersects CD produced at E

=> angle AED + angle ABC = 180  ... (1)  (Sum of opposite angles of cyclic quadrilaterals is 180)
=> angle ADE + angle ADC = 180  ... (2)  (linear pair)
=> angle ABC = angle ADC  ... ........(3)  (opposite angles of parallelogram are equal)
From (1) and (2)
=> angle AED + angle ABC = angle ADE + angle ADC
=> angle AED = angle ADE  (using (3))
In triangle AED,
=> AD = AE   (equal sides have equal angles opposite to them)
Hence Proved
Answered by Expert 23rd February 2013, 10:05 PM
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