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. *)
(** * Proposition 11 If two circles touch one another internally, and their centers are taken, the straight line joining their centers, being produced, will fall on the point of contact of the circles. *)
(** * Proposition 12 If two circles touch one another externally, the straight line joining their centers will pass through the point of contact. *)
(** In our formalization we do not need to distinguish between the two kinds of tangency. *)

Lemma prop_11_12 : forall A B C D X,
TangentCC A B C D ->
OnCircle X A B ->
OnCircle X C D ->
Col X A C.