(** Given a circle, there is a point strictly outside the circle. *)

(* Specifications: *)
Lemma outcs_exists : forall O P, exists Q, OutCircleS Q O P.
(** Given a circle of center O and a ray OX, there is a point on the ray which is also strictly outside the circle. *)
Lemma outcs_exists1 : forall O P X, X <> O -> exists Q, Out O X Q /\ OutCircleS Q O P.
(** Given a circle there is a point which is strictly inside. *)
Lemma incs_exists : forall O P, O <> P -> exists Q, InCircleS Q O P.
(** Given a circle of center O and a ray OX, there is a point on the ray which is also strictly inside the circle. *)
Lemma incs_exists1 : forall O P X, X <> O -> P <> O -> exists Q, Out O X Q /\ InCircleS Q O P.
(** Given a circle of center O and a ray OX, there is a point on the ray which is also on the circle. *)
Lemma onc_exists : forall O P X,  X <> O -> O <> P -> exists Q, OnCircle Q O P /\ Out O X Q.
(** Given a circle of center O and a line OX, O is between two points of the line which are also on the circle. *)
Lemma diam_points : forall O P X, exists Q1 Q2,
Bet Q1 O Q2 /\ Col Q1 Q2 X /\ OnCircle Q1 O P /\ OnCircle Q2 O P.