(** This following result is very important, it shows the existence of a midpoint. The proof is involved because we are not using continuity axioms. *)
(** This corresponds to l8_22 in Tarski's book. *)

Lemma l8_22 : forall A B P R X ,
A <> B -> A <> P ->
Per B A P -> Per A B R ->
Cong A P B R -> Col A B X -> Bet P X R ->
Cong A R P B /\ Midpoint X A B /\ Midpoint X P R.