Section Tangency.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** Euclid Book III, Prop 11 and Prop 12 We do not need to distinguish between internal or external tangency. *)
(** If two circles are tangent, the common point is on the line joining the centers. *)
(** A line going through the center is not tangent to the circle. *)
(** Every point on the tangent different from the point of tangency is strictly outside the circle. *)
(** If line AB is tangent to a circle of center O at a point T, then OT is perpendicular to AB. This is Euclid Book III, Prop 18 *)
(** AB is tangent to the circle (O,P) iff they intersect at a point X such that AB is perpendicular to OX. *)

Lemma tangency_chara2 : forall A B O P Q,
OnCircle Q O P -> Col Q A B ->
((forall X, Col A B X -> X = Q \/ OutCircleS X O P) <-> Tangent A B O P).