Ltac double A B A' :=
   assert (mp:= symmetric_point_construction A B);
   elim mp; intros A' ; intro; clear mp.

Lemma l8_20_2 : forall A B C C' D P, Per A B C -> Midpoint P C' D -> Midpoint A C' C -> Midpoint B D C -> B<>C -> A<>P.