(** The following lemmas express the reverse triangle inequality only with predicates from Tarski : BC >= |AC - AB|, and BC = |AC - AB| <-> (A = B \/ A = C \/ Out A B C) *)

(* Specifications: *)
Lemma triangle_strict_reverse_inequality : forall A B C D,
Out A B D -> Cong A C A D -> ~ Out A B C -> Lt B D B C.
Lemma triangle_reverse_inequality : forall A B C D, Out A B D -> Cong A C A D -> Le B D B C.
(** This is half of Euclid Book I, Proposition 21: if D is inside the triangle ABC then BAC < BDC. *)
Lemma os3__lta : forall A B C D, OS A B C D -> OS B C A D -> OS A C B D ->
LtA B A C B D C.