proposition: Lemma triangle_mid_par_flat_cong : forall A B C P Q R, B <> A -> C <> A -> C <> B -> Col A B C -> Midpoint P B C -> Midpoint Q A C -> Midpoint R A B -> Par A B Q P /\ Par A C P R /\ Par B C Q R /\ Cong A R P Q /\ Cong B R P Q /\ Cong A Q P R /\ Cong C Q P R /\ Cong B P Q R /\ Cong C P Q R.
proof: Proof. intros. permutation_intro_in_hyps. assert (Par A B Q P /\ Cong A R P Q /\ Cong B R P Q) by (apply triangle_mid_par_flat_cong_aux with C; assumption). assert (Par A C R P /\ Cong A Q P R /\ Cong C Q P R) by (apply triangle_mid_par_flat_cong_aux with B; Col). assert (Par C B Q R /\ Cong C P R Q /\ Cong B P R Q) by (apply triangle_mid_par_flat_cong_aux with A; Col). spliter. repeat split; Cong; Par. Qed.
imports: GeoCoq.Tarski_dev.Annexes.quadrilaterals_inter_dec