Ltac CopR :=
 let tpoint := constr:(Tpoint) in
 let col := constr:(Col) in
 let cop := constr:(Coplanar) in
   assert_ncols; Cop_refl tpoint col cop.

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.