Lemma l_3_45 : forall A B C E F G A1 B1 C1 r, is_midpoint E A C -> on_perp_d B1 E A r -> is_midpoint F B C -> on_perp_d A1 F C r -> is_midpoint G A B -> on_perp_d C1 G B (0-r) -> parallel A1 C1 C B1.