(** Euclid Book III Prop 23: On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. *)

Lemma conga_cop_onc6_os__eq : forall A B C D O P O' P',
OnCircle A O P -> OnCircle B O P -> OnCircle C O P -> Coplanar A B C O ->
OnCircle A O' P' -> OnCircle B O' P' -> OnCircle D O' P' -> Coplanar A B D O' ->
OS A B C D -> CongA A C B A D B ->
(O = O' /\ Cong O P O' P').