Lemma proposition_47 : forall A B C D E F G H K, Triangle A B C -> Per B A C -> SQ A B F G -> TS G B A C -> SQ A C K H -> TS H C A B -> SQ B C E D -> TS D C B A -> exists X Y, PG B X Y D /\ BetS B X C /\ PG X C E Y /\ BetS D Y E /\ EF A B F G B X Y D /\ EF A C K H X C E Y.