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. *)

Lemma inscribed_angle : forall O P A B C,
OnCircle A O P -> OnCircle B O P -> OnCircle C O P -> OS A B O C ->
gHalfA A C B A O B.