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.

Answer :


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),


⇒∠EAD + DAF = 90°


⇒∠EAF = 90°


EAF is the angle in a semicircle.


EF is the diameter of the circle.


Hence proved.


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