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. *)
(** Given a circle, there is a point strictly outside the circle. *)
(** Given a circle of center O and a ray OX, there is a point on the ray which is also strictly outside the circle. *)
(** Given a circle there is a point which is strictly inside. *)
(** Given a circle of center O and a ray OX, there is a point on the ray which is also strictly inside the circle. *)

Lemma incs_exists1 : forall O P X, X <> O -> P <> O -> exists Q, Out O X Q /\ InCircleS Q O P.