Q5 of 33 Page 57

In the picture, bisectors of adjacent angles of the quadrilateral ABCD intersect at P, Q, R, S.


Prove that PQRS is a cyclic quadrilateral.

In the given figure,


∠ASD = ∠PSR [Vertically opposite angles]…[1]


In ΔASD


∠ASD + ∠ADS + ∠DAS = 180° [By angle sum property]


Also, As AS and DS are angle bisectors, therefore


and



Using these and from equation [1] we have,



…[2]


Similarly,


…[3]


Adding [2] and [3]



Also, ∠A + ∠B + ∠C + ∠D = 360° [By angle sum property of quadrilateral]



⇒ ∠PSR + ∠PQR = 180°


⇒ PQRS is a cyclic quadrilateral.


[As in a cyclic quadrilateral, sum of any pair of opposite angles is 180°].


More from this chapter

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Prove that a non-isosceles trapezium is not cyclic.

 6

The two circles below intersect at P, Q and lines through these points meet the circles at A, B, C, D. Prove that if AC and BD are of equal length, then ABCD is a cyclic quadrilateral.

 6

In the picture, the circles on the left and right intersect the middle circle at P, Q, R, S; the lines joining them meet the left and right circles at A, B, C, D. Prove that ABCD is a cyclic quadrilateral.