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. *)
(** * Proposition 3 If in a circle a straight line passing through the center bisects a straight line not passing through the center, it also cuts it at right angles; and if it cut it at right angles, it also bisects it. *)
(** * Proposition 4 If in a circle two straight lines cut one another which are not through the center, they do not bisect one another. *)
(** * Proposition 5 If two circles cut one another, they will not have the same center. *)
(** * Proposition 6 If two circles touch one another, they will not have the same center. *)
(** * Proposition 9 If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, the point taken is the center of the circle. *)

Lemma prop_9 : forall O P X A B C, A <> B -> A <> C -> B <> C ->
OnCircle A O P ->
OnCircle B O P ->
OnCircle C O P ->
Cong X A X B ->
Cong X A X C ->
X = O.