Section HalfAngle.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Definition HalfA P A O B := ~ Bet A O B /\ InAngle P A O B /\ CongA A O P B O P.
(* Lemma lta_nbet__ncol : forall A B C X Y Z, ~ Bet X Y Z -> LtA A B C X Y Z -> ~ Col X Y Z. *)
Definition gHalfA A' O' B' A O B := exists P, HalfA P A O B /\ CongA A' O' B' A O P.
(** Given two angles a and b, a/2 = b/2 -> a = b *)
(** Given two angles a and b, a = b -> a/2 = b/2 *)
(** Unicity of the double angle *)
(** Unicity of the half angle *)
(** CongA preserves gHalfA *)
(** Out preserves gHalfA *)
(** Given two angles a and b such that a/2 + b/2 is acute, a/2 + b/2 = (a+b)/2 *)
(** Given two angles a and b, a/2 - b/2 = (a-b)/2 *)

Lemma acute_ghalfa2_sams_suma2__ghalfa456 : forall A B C D E F X Y Z A' B' C' D' E' F' X' Y' Z',
SAMS A' B' C' D' E' F' -> Acute D E F ->
SumA A B C D E F X Y Z -> SumA A' B' C' D' E' F' X' Y' Z' ->
gHalfA A B C A' B' C' -> gHalfA X Y Z X' Y' Z' ->
gHalfA D E F D' E' F'.