unfold orthogonal, double in |- *; intros u v H.
apply transitive with (plus (cons v u) pi); auto.
apply regulier with (a := cons u v) (c := cons u v); auto.
apply transitive with pi; auto.
apply transitive with (plus (plus (cons u v) (cons v u)) pi); auto.
apply transitive with (plus zero pi); auto.
apply compatible; auto.
Qed.
Hint Resolve critere_orthogonal critere_orthogonal_reciproque.
Definition bissectrice (I A B C : PO) :=
R (cons (vec A B) (vec A I)) (cons (vec A I) (vec A C)).

(* Specifications: *)
Lemma bissectrice_double :
forall I A B C : PO,
bissectrice I A B C ->
R (double (cons (vec A I) (vec A C))) (cons (vec A B) (vec A C)).