Definition on_perp (A' B C : Point) : Prop := B <> C /\ perp A' B B C /\ S B C A' <> 0.Definition on_perp_d (Y U V : Point)  (lambda : F) : Prop := 
     U <> V  /\ perp Y U U V /\ lambda * Py U V U = (2+2) * S U V Y /\ lambda <> 0.Definition on_inter_line_perp (Y R U V P Q : Point) : Prop :=
  Col Y U V /\ perp Y R P Q /\ ~ perp P Q U V.

Lemma on_line_d_iff_on_parallel_d : forall A B C r, on_line_d A B C r <-> on_parallel_d A B C B (0-r).