(** * Euclid's Elements  Book III *)
Require Export GeoCoq.Tarski_dev.Annexes.tangency.
Require Export GeoCoq.Tarski_dev.Annexes.inscribed_angle.
Section Book_3.
Context `{TE:Tarski_2D}.
(** * Proposition 1 To find the center of a given circle. *)
(** We do not formalize this proposition, because in our formalization circles are given by their center and one point on the circle. *)
(** * Proposition 2 If on the circumference of a circle two points are taken at random, the straight line joining the points will fall within the circle. *)

Lemma prop_2 : forall O P U V X,
X <> U -> X <> V ->
Bet U X V ->
OnCircle U O P ->
OnCircle V O P ->
InCircleS X O P.