Question

ABCD is a parallelogram. A circle through vertics A and B meets side BC at the point P and side AD at point Q. Show that quadrilateral PCDQ is cyclic..
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Answer 1

Given ABCD is a parallelogram. A circle whose centre O passes through A, B is so drawn that it intersect AD at P and BC at Q To prove Points P, Q, C and D are con-cyclic.

Construction Join PQ

Proof ∠1 = ∠A [exterior angle property of cyclic quadrilateral]

But ∠A = ∠C [opposite angles of a parallelogram]

∴ ∠1 = ∠C ,..(i)

But ∠C+ ∠D = 180° [sum of cointerior angles on same side is 180°]

⇒ ∠1+ ∠D = 180° [from Eq. (i)]

Thus, the quadrilateral QCDP is cyclic.

So, the points P, Q, C and D are con-cyclic. Hence proved.

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