(* Lemma triangle_circumscription_aux : forall A B C P, Cong A P B P -> Cong A P C P -> exists CC, Cong A CC B CC /\ Cong A CC C CC /\ Coplanar A B C CC. *)

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.