(** Euclid Book III Prop 3 (two lemmas). If a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then it also bisects it. *)
(** The line from the center of a circle to the midpoint of a chord is perpendicular to the chord. *)

Lemma mid_onc2__perp : forall O P A B X,
O <> X -> A <> B -> OnCircle A O P -> OnCircle B O P -> Midpoint X A B -> Perp O X A B.