(** Euclid Book III Prop 9. If a point is taken within a circle, and more than two equal straight lines fall from the point on the circle, then the point taken is the center of the circle. *)

Lemma onc3_mid2__ncol : forall O P A B C A' B',
O <> P ->
OnCircle A O P -> OnCircle B O P -> OnCircle C O P ->
Midpoint A' A C -> Midpoint B' B C -> ~ Col A B C ->
~ Col O A' B' \/ A' = O \/ B' = O.