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  • ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C and D are concyclic.

Answers (1)

Given: ABCD is a parallelogram. A circle whose centre O passes through A, B has intersected AD at P and BC at Q.

To Prove: Points P, Q, C and D are concyclic. 

Proof:

Join PQ

We know that sum of opposite angles of a cyclic quadrilateral is 180°      

\angleA + \anglePQB = 180°

Also, \anglePQB + \anglePQC = 180°   (linear pair)

Comparing the above two equations we get

\anglePQC = \angleA

Let \anglePQC = \angle1

Then, \angleA = \angle1

Also, \angleA = \angleC  (Opposite angles of a parallelogram are equal)

So, \angle1 = \angleC

But, \angleC + \angleD = 180°  (As ABCD is a parallelogram)

Hence, \angle1 + \angleD = 180°

Therefore P, Q, C and D are concyclic

Hence proved

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