proposition: Lemma midpoint_preserves_per : forall A B C A1 B1 C1 M, Per A B C -> Midpoint M A A1 -> Midpoint M B B1 -> Midpoint M C C1 -> Per A1 B1 C1.
proof: Proof. intros. unfold Per in *. ex_and H C'. double C' M C1'. exists C1'. split. eapply symmetry_preserves_midpoint. apply H2. apply H1. apply H4. assumption. eapply l7_16. apply H0. apply H2. apply H0. apply H4. assumption. Qed.
imports: GeoCoq.Tarski_dev.Ch09_plane GeoCoq.Tarski_dev.Tactics.CoincR_for_cop