If the bisector of angle W and angle Y of a cyclic quadrilateral WXYZ meet at A and B respectively then prove that AB is the diameter of the circle. ?

1 Answer
Apr 28, 2018

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Given

that the bisectors of angle W and angle Y of a cyclic quadrilateral WXYZ meet the circle at A and B respectively.
AandB are joined.

RTP

To prove that AB is the diameter of the circle.

Construction

AandY are joined.

Proof

Sum of opposite angles of a cyclic quadrilateral being 180 we have for cyclic quadrilateral WXYZ

XWZ+XYZ=180

12XWZ+12XYZ=12×180

XWA+XYB=90 [ since WA and YB are bisectors of XWZandXYZ respectively.]

Bur XWA=XYA, being the angles on same arc AX

So We have

XYA+XYB=90

AYB=90

This means AB must be diameter of the circle.