The bisectors of opposite angles `/_ P` and `/_ R` of a cyclic quadrilateral PQRS intersect the correspondig circle at A and B respectively.
Prove that AB is a diameter of the circle.

Explanation

GIVEN A cyclic quad. PQRS in which PA and RB are the bisectors of `/_P` and `/_ R` respectively
TO PROVE AB is a diameter of the circle which passes through the points P,Q,R and S.
CONSTRUCTION Join PB and BS.
PROOF SincePQRS is a cyclic quadrilateral , we have
`/_ QPS + /_ QRS = 180^(@)`
`implies (1)/(2) /_ QPS +(1)/(2) /_QRS = 90^(@)`
`implies /_ APS +/_ BRS = 90^(@)`
`implies /_ APS +/_BPS =90^(@)`
`[ :' /_ BRS = /_ BPS , /_ s` in the same segment ]
`implies /_ APB = 90^(@) ` `[:' /_ APS+/_ BPS implies APB]`
`implies /_ APB ` is in a semicircle
`implies` AB is a diameter of the circle which passes through the points P,Q,R and S
Hence, AB isa diameter of the circle.