Proof with finish.
intros.
permutation_intro_in_hyps.
assert (Par_strict A B Q P /\ Cong A R P Q /\ Cong B R P Q)
by (apply triangle_mid_par_strict_cong_aux with C; assumption).
assert (Par_strict A C R P /\ Cong A Q P R /\ Cong C Q P R)
by (apply triangle_mid_par_strict_cong_aux with B; Col).
assert (Par_strict C B Q R /\ Cong C P R Q /\ Cong B P R Q)
by (apply triangle_mid_par_strict_cong_aux with A; Col).
spliter.
split...
split...
split...
repeat split...
Qed.

Lemma triangle_mid_par_flat_cong_aux : forall A B C P Q R,
B <> A ->
Col A B C ->
Midpoint P B C ->
Midpoint Q A C ->
Midpoint R A B ->
Par A B Q P /\ Cong A R P Q /\ Cong B R P Q.