Definition HalfA P A O B := ~ Bet A O B /\ InAngle P A O B /\ CongA A O P B O P.

Lemma halfa_chara1 : forall P A O B, HalfA P A O B -> exists A', exists M, Cong O A' O A /\ Out O A' B /\ Midpoint M A A' /\ Out O M P.