Section Dedekind_circle_circle.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** This proof is inspired by Franz Rothe's proof of Theorem 8.5 in Several topics from geometry *)

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.