Section Circle_2D.
Context `{T2D:Tarski_2D}.
(** The center of a circle belongs to the perpendicular bisector of each chord *)
(** Euclid Book III Prop 4. If in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another. *)
(** Euclid Book III Prop 9. If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle. *)

Lemma bet_cong_onc3_cases : forall O P A X Y T,
T <> O -> Bet A O T -> Cong T X T Y ->
OnCircle A O P  -> OnCircle X O P  -> OnCircle Y O P ->
X = Y \/ ReflectL X Y O A.