Require Export GeoCoq.Highschool.bisector.
Require Export GeoCoq.Tarski_dev.Ch13_1.
Section InCenter.
Context `{TE:Tarski_euclidean}.
Definition is_incenter I A B C :=
~ Col A B C /\ CongA B A I I A C /\ CongA A B I I B C /\ CongA A C I I C B.
(** Proof of the existence of the incenter of a triangle. *)

(* Specifications: *)
Lemma incenter_exists : forall A B C, ~ Col A B C -> exists I, is_incenter I A B C.
Lemma incenter_permut132 : forall A B C I, is_incenter I A B C -> is_incenter I A C B.
Lemma incenter_permut213 : forall A B C I, is_incenter I A B C -> is_incenter I B A C.
Lemma incenter_permut231 : forall A B C I, is_incenter I A B C -> is_incenter I B C A.
Lemma incenter_permut312 : forall A B C I, is_incenter I A B C -> is_incenter I C A B.
Lemma incenter_permut321 : forall A B C I, is_incenter I A B C -> is_incenter I C B A.
Lemma incenter_dec : forall A B C I, is_incenter I A B C \/ ~ is_incenter I A B C.
End InCenter.