Lemma circle_circle_aux : (forall A B C D P Q, OnCircle P C D -> OnCircle Q C D -> InCircleS P A B -> OutCircleS Q A B -> OS A C P Q \/ (Col P A C /\ ~ Col Q A C) \/ (~ Col P A C /\ Col Q A C ) -> exists Z : Tpoint, OnCircle Z A B /\ OnCircle Z C D) -> circle_circle.