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

Lemma Perp_in_cases : forall X A B C D, Perp_at X A B C D \/ Perp_at X B A C D \/ Perp_at X A B D C \/ Perp_at X B A D C \/ Perp_at X C D A B \/ Perp_at X C D B A \/ Perp_at X D C A B \/ Perp_at X D C B A -> Perp_at X A B C D.