unfold colineaire in |- *; intros u v w t H H0.
apply transitive with (plus (cons v u) (cons u t)); auto.
apply transitive with (plus (cons u w) (cons u t)); auto.
apply compatible; auto.
apply regulier with (a := cons u t) (c := cons u t); auto.
apply transitive with (cons u w); auto.
apply transitive with (plus (cons u t) (plus (cons u t) (cons u w))); auto.
apply compatible; auto.
apply transitive with (plus (plus (cons u t) (cons u t)) (cons u w)); auto.
apply transitive with (plus zero (cons u w)); auto.
apply compatible; auto.
Qed.
Hint Resolve bissectrice_droite.
Definition milieu (I B C : PO) :=
colineaire (vec B I) (vec B C) /\
(forall A : PO, isocele A B C -> bissectrice I A B C).

(* Specifications: *)
Axiom existence_milieu : forall B C : PO, exists I : PO, milieu I B C.
Lemma milieu_isocele :
forall I A B C : PO, isocele A B C -> milieu I B C -> bissectrice I A B C.