Q. 28

# If the bisectors of the opposite angles of a cyclic quadrilateral ABCD intersects the circumscribed circle of this quadrilateral at points P and Q, then prove that PQ is a diameter of this circle.

Given ABCD is a cyclic quadrilateral. AP and CQ are bisectors of A and C respectively.

We have to prove that PQ is the diameter of the circle.

Construction:

Join AF and FD

Proof:

We know that in a cyclic quadrilateral, opposite angles are supplementary.

⇒∠A + C = 180°

1/2 A + 1/2 C = 90°

⇒∠EAD + DCF = 90° … (1)

We know that angles in the same segment are equal.

⇒∠DCF = DAF … (2)

From (1) and (2),

⇒∠EAF = 90°

EAF is the angle in a semicircle.

EF is the diameter of the circle.

Hence proved.

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