Lemma triangle_circumscription_implies_tarski_s_euclid_aux : forall A B C D T X Y Z M1 Z1 M2 Z2, triangle_circumscription_principle -> A <> B -> A <> C -> A <> D -> A <> T -> A <> Y -> B <> C -> B <> D -> B <> T -> B <> Y -> C <> D -> C <> T -> C <> Y -> D <> T -> T <> X -> T <> Y -> X <> Y -> Y <> Z1 -> Y <> Z2 -> ~ Col A B C -> Col A B M1 -> Col A C M2 -> Bet A D T -> ~ Col B C T -> Bet B D C -> Col T Y Z -> Bet Y T X -> Bet Y M1 Z1 -> Bet Y M2 Z2 -> Cong Y T T X -> Cong Y M1 M1 Z1 -> Cong Y M2 M2 Z2 -> Perp B C T Z -> Perp A B Y Z1 -> Perp A C Y Z2 -> exists x, exists y, Bet A B x /\ Bet A C y /\ Bet x T y.