Question

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. ?

Answer 1

Given

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

RTP

To prove that AB is the diameter of the circle.

Construction

`A and Y` are joined.

Proof

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

`angle XWZ+angle XYZ=180^@`

`=>1/2angle XWZ+1/2angle XYZ=1/2xx180^@`

`=>angle XWA+angle XYB=90^@` [ since WA and YB are bisectors of `angle XWZ and angle XYZ` respectively.]

Bur `angle XWA=angleXYA`, being the angles on same arc `AX`

So We have

`angle XYA+angle XYB=90^@`

`=>angle AYB=90^@`

This means `AB` must be diameter of the circle.