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 Prop_7_8_uniqueness : forall O P A X Y Z T, T <> O -> X <> Y ->
Bet A O T -> Cong T X T Y -> Cong T X T Z ->
OnCircle A O P -> OnCircle X O P -> OnCircle Y O P -> OnCircle Z O P ->
Z = X \/ Z = Y.