Section Circle_2.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** If a point is strictly inside a segment of a disk, it is strictly inside the circle. *)
(** If A and B are two points inside the circle, then all points on the segment AB are also in the circle, i.e. a circle is a convex figure. *)
(** Given two points U and V on a circle, the points of the line UV which are inside the circle are between U and V. *)
(** Given two points U and V on a circle, all points of line UV which are outside the segment UV are outside the circle. *)
(** Given two points U and V inside a circle, all points of line UV which are outside the circle are outside the segment UV. *)
(** If the center of a circle belongs to a chord, then it is the midpoint of the chord. *)
(** Given a point U on a circle and a point P inside the circle, there is a point V such as UV is a chord of the circle going through P. *)

Lemma chord_completion : forall A B U P, OnCircle U A B -> InCircle P A B ->
exists V, OnCircle V A B /\ Bet U P V.