Require Export GeoCoq.Meta_theory.Parallel_postulates.SPP_ID.
Require Export GeoCoq.Meta_theory.Parallel_postulates.TCP_tarski.
Require Export GeoCoq.Meta_theory.Parallel_postulates.alternate_interior_angles_proclus.
Require Export GeoCoq.Meta_theory.Continuity.angle_archimedes.
Require Export GeoCoq.Meta_theory.Continuity.archimedes.
Require Export GeoCoq.Meta_theory.Continuity.aristotle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.euclid_5_original_euclid.
Require Export GeoCoq.Meta_theory.Parallel_postulates.existential_triangle_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.par_perp_perp_TCP.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_alternate_interior_angles.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_bis_playfair.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_universal_posidonius_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.proclus_SPP.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rah_triangle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.tarski_euclid.
Require Export GeoCoq.Meta_theory.Parallel_postulates.triangle_playfair_bis.
Require Export GeoCoq.Meta_theory.Parallel_postulates.triangle_existential_triangle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.universal_posidonius_postulate_par_perp_perp.
Require Export GeoCoq.Tarski_dev.Annexes.defect.
Section Legendre.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

Lemma legendre_aux1 : forall A B C B' C',
~ Col A B C -> Out A B B' -> Out A C C' ->
exists D', InAngle D' B A C /\ CongA A C' B' C' B' D' /\
Cong A C' B' D' /\ TS B' C' A D'.