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. *)
(** Given a circle of center O and a ray OX, there is a point on the ray which is also on the circle. *)
(** Given a circle of center O and a line OX, O is between two points of the line which are also on the circle. *)
(** The symmetric of a point on a circle relative to the center is also on the circle. *)
(** The middle of a chord together with the center of the circle and one end of the chord form a right angle *)
(** Euclid Book III Prop 3 (two lemmas). If a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then it also bisects it. *)
(** The line from the center of a circle to the midpoint of a chord is perpendicular to the chord. *)
(** If a line passing through the center of a circle is perpendicular to a chord, then they intersect at the middle of the chord *)
(** If two circles intersect at a point which is not on the line joining the center, then they intersect on any half-plane delimited by that line. *)
(** If two circles intersect, then they intersect on any plane containing the centers *)
(** A circle does not cut a line at more than two points. *)
(** The midpoint of a chord is strictly inside the circle. *)
(** A point is either strictly inside, on or strictly outside a circle. *)
(** If a point is inside a circle, then it lies on a radius. *)
(** Euclid Book III Prop 14 Equal straight lines in a circle are equally distant from the center, and those which are equally distant from the center equal one another. *)
(** Prop 7   **)
(** Two circles are equal if and only if they have the same center and the same radius *)
(** Two circles are distinct if and only if there is a point belonging to the first and not to the second *)
(** The EqC predicate is an equivalence relation on ordered pairs of points *)
(** If two circles have three common points, then they are equal *)
(** If four coplanar points belong to a same sphere, then they belong to a same circle. This lemma justifies our definition of Concyclic. *)

Lemma concyclic_aux : forall A B C D, Concyclic A B C D -> exists O P,
OnCircle A O P /\ OnCircle B O P /\ OnCircle C O P /\ OnCircle D O P /\ Coplanar A B C O.