Definition gHalfA A' O' B' A O B := exists P, HalfA P A O B /\ CongA A' O' B' A O P.

Lemma acute_ghalfa2_sams_suma2__ghalfa123 : 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 A B C -> SumA A B C D E F X Y Z -> SumA A' B' C' D' E' F' X' Y' Z' -> gHalfA D E F D' E' F' -> gHalfA X Y Z X' Y' Z' -> gHalfA A B C A' B' C'.