This is your construction - get 14A and 14B ... and you can not get it.
It does not prove it can not be construcated by other ways.
Suggestion:
Read 康MMprevious post and follows.康M guess, this construction of 7 is unique. Try to add one point on it, to see what happen, then you got some feeling of case 8. But, you can not prove this construction is unique. Key point: all points are on two unit circle A and D, that is generic.