Section Inscribed_angle.
Context `{TE:Tarski_euclidean}.
(** The sum of the angles of a triangle is the flat angle. *)
(** If A, B and C are points on a circle where the line AB is a diameter of the circle, then the angle ACB is a right angle. *)
(** In a right triangle, the midpoint of the hypotenuse is the circumcenter. *)
(** If the angle ACB is inscribed in a circle of center O and C, O lie on the same side of AB, then this angle is acute. *)
(** Euclid Book III Prop 20: In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base. *)
(** In a circle the angle at the centre is double of the angle at the circumference. *)
(** If two angles ACB and ADB are inscribed in the same circle, then they are either congruent or supplementary. *)
(** If the angle ACB is inscribed in a circle of center O and C, O lie on opposite sides of AB, then this angle is obtuse. *)
(** Euclid Book III Prop 21: In a circle the angles in the same segment are equal to one another. *)
(** Euclid Book III Prop 22: The opposite angles of quadrilaterals in circles are equal to two right angles. *)
(** If the angle ACB is acute and inscribed in a circle of center O, then C and O lie on the same side of AB. *)
(** If the angle ACB is obtuse and inscribed in a circle of center O, then C and O lie on opposite sides of AB. *)
(** If the angles ACB and ADB are congruent and inscribed in the same circle, then C and D lie on the same side of AB. *)
(** If the angles ACB and ADB are supplementary and inscribed in the same circle, then C and D lie on opposite sides of AB. *)
(** Non degenerated triangles can be circumscribed. *)

Lemma triangle_circumscription : forall A B C, ~ Col A B C ->
exists CC : Tpoint, Cong A CC B CC /\ Cong A CC C CC /\ Coplanar A B C CC.