(** 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 chords_midpoints_col_par : forall O P A M B C N D,
O <> P ->
OnCircle A O P -> OnCircle B O P ->
OnCircle C O P -> OnCircle D O P ->
Midpoint M A B -> Midpoint N C D ->
Col O M N -> ~ Col O A B -> ~ Col O C D -> Par A B C D.