Question

20. Prove that the bisectors of the angles formed by producing the opposite sides of a cyclic quadrilateral
provided that they are not parallel), intersect at right angles.​

Answer 1

ABCD is a cyclic quadrilateral. The bisectors of its opposite angles A and C intersect the circle circumscribing at the points P and Q respectively.

⇒  ∠A+∠C=180o             [ Opposite angles of a cyclic quadrilateral are supplementary ]

⇒  21∠A+21∠C=90o

⇒  ∠PAB+∠BCQ=90o

But ∠BCQ=∠BAQ             [ Angles in the segment of a circle are equal ]

∴  ∠PAB+∠BAQ=90o

⇒  ∠PAQ=90o

⇒  But ∠PAQ is in semicircle.

∴  PQ is diameter of the circle.

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