Proof with finish.
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...
Qed.

Lemma triangle_mid_par : forall A B C P Q,
A <> B ->
Midpoint P B C ->
Midpoint Q A C ->
Par A B Q P.