proposition: Lemma par_cong_mid : forall A B A' B', Par A B A' B' -> Cong A B A' B' -> exists M, Midpoint M A A' /\ Midpoint M B B' \/ Midpoint M A B' /\ Midpoint M B A'.
proof: Proof. intros. induction H. apply par_strict_cong_mid; try assumption. apply col_cong_mid. unfold Par. right. assumption. intro. unfold Par_strict in H1. spliter. apply H2. exists A. split; Col. assumption. Qed.
imports: GeoCoq.Tarski_dev.Ch12_parallel_inter_dec GeoCoq.Tarski_dev.Annexes.Tagged_predicates