OK, a few things:
1) I don't think it matters where T and S are, so long as T lies on the line OQ (line, not line segment), and S lies on the line RO (again, line, not line segment), and that OT = OS

However, i don't think it can be done because of this case:

1) Construct a circle centre O
2) choose a point P outside O, and join the 2 chords PR and PS
3) Join RO and SO.
4) Place T, between the circumference of the circle the 'other' side of Q, and O (i.e. TOQ is the smaller than diameter, but bigger than the radius.)
5) Place S on the line segment RO, so that OS = OT.
6) RQST are definitely not cyclic, as S is inside RQT