We know that if two chords of two circles intersect at the radical axis, then the four end points of those chords are concyclic. But I do not understand how the case when the two chords are collinear is handled:
What is the circle defined by A,B,C and D?
The theorem states that if two non-collinear chords of two circles intersect at the radical axis, then the four end points of those chords are concyclic. Collinear chords will have inifinite points of intersection, so it does not make any sense to define this theorem for them.