Require Export GeoCoq.Tarski_dev.Ch13_1.
Require Export GeoCoq.Highschool.triangles.
Section Bisector.
Context `{TE:Tarski_euclidean}.
(** Existence of the interior bisector of an angle. *)

(* Specifications: *)
Lemma bisector_existence : forall A B C,  A <> B -> B <> C ->
exists E,  InAngle E A B C /\ CongA A B E E B C.
Lemma not_col_bfoot_not_equality : forall A B C I H1 H2, ~ Col A B C -> Coplanar A B C I ->
Col A B H1 -> Col B C H2 -> CongA A B I I B C->
Perp A B I H1 -> Perp B C I H2 -> H1 <> B /\ H2 <> B.
Lemma bisector_foot_out_out : forall A B C I H1 H2, ~ Col A B C ->
Coplanar A B C I -> Col A B H1 -> Col B C H2-> CongA A B I I B C->
Perp A B I H1 -> Perp B C I H2 -> (Out B H1 A -> Out B H2 C).
Lemma not_col_efoot_not_equality : forall A B C I H1 H2, ~ Col A B C -> Coplanar A B C I ->
Col A B H1 -> Col B C H2-> Cong I H1 I H2->
Perp A B I H1 -> Perp B C I H2 -> H1 <> B /\ H2 <> B.