Section triangle_playfair_bis.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma legendre_aux1 :
greenberg_s_axiom ->
triangle_postulate ->
forall A1 A2 B1 B2 C1 C2 P,
Perp2 A1 A2 B1 B2 P -> ~ Col A1 A2 P -> Col P B1 B2 -> Coplanar A1 A2 B1 B2 ->
Par A1 A2 C1 C2 -> Col P C1 C2 -> ~ TS B1 B2 A1 C1 ->
Col C1 B1 B2.
Lemma legendre_aux2 :
greenberg_s_axiom ->
triangle_postulate ->
forall A1 A2 B1 B2 C1 C2 P,
Perp2 A1 A2 B1 B2 P -> ~ Col A1 A2 P -> Col P B1 B2 -> Coplanar A1 A2 B1 B2 ->
Par A1 A2 C1 C2 -> Col P C1 C2 ->
Col C1 B1 B2. (* "half" of playfair_bis *)
Lemma triangle__playfair_bis :
greenberg_s_axiom ->
triangle_postulate ->
alternative_playfair_s_postulate.