proposition: Lemma vecteur_milieu : forall A B M : PO, vec A M = mult_PP (/ 2) (vec A B) :>PP -> M = milieu A B :>PO.
proof: Proof. intros; unfold milieu in |- *. rewrite (colineaire_barycentre (k:=/ 2) (A:=A) (B:=B) (C:=M)); auto. cut (1 + - / 2 = / 2); intros; auto with real. rewrite H0. RReplace (/ 2) (/ 2 * 1). apply homogene_barycentre; auto with real. Qed.