(** Predicates to compare segment lengths. *)
  
Definition le := fun A B C D =>
   exists y, Bet C y D /\ Cong A B C y.Definition ge := fun A B C D => le C D A B.

Lemma segment_construction_2 : forall A Q B C, A<>Q -> exists X, (Bet Q A X \/ Bet Q X A) /\ Cong Q X B C.