Section T11_2.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** This is exterior angle theorem *)
(** In an isosceles triangle the two base angles are equal. This is Euclid: Book 1, Proposition 5. *)
(** This is Euclid: Book 1, Proposition 18 *)
(** If the base angles are equal, then the triangle is isosceles *)
(** This is Euclid Book I, Proposition 19 *)
(** This is SAS congruence. *)
(** This is ASA congruence. *)
(** This is AAS congruence. *)
(** This is SSS congruence. *)
(** This is SSA congruence with a restriction *)
(** This is SSA congruence with an obtuse angle *)
(** This is SSA congruence with a right angle *)
(** The following lemmas express the triangle inequality only with predicates from Tarski : AC <= AB + BC, and AC = AB + BC <-> Bet A B C *)
(** The following lemmas express the reverse triangle inequality only with predicates from Tarski : BC >= |AC - AB|, and BC = |AC - AB| <-> (A = B \/ A = C \/ Out A B C) *)
(** This is half of Euclid Book I, Proposition 21: if D is inside the triangle ABC then BAC < BDC. *)
(** This is Theorem 18.17 from Martin *)
(** This is Euclid Book I, Proposition 24. *)

Lemma angle_bisector : forall A B C, A <> B -> C <> B ->
exists P, InAngle P A B C /\ CongA P B A P B C.