Section Book_3.
Context `{TE:Tarski_2D}.
(** * Proposition 1 To find the center of a given circle. *)
(** We do not formalize this proposition, because in our formalization circles are given by their center and one point on the circle. *)
(** * Proposition 2 If on the circumference of a circle two points are taken at random, the straight line joining the points will fall within the circle. *)
(** * Proposition 3 If in a circle a straight line passing through the center bisects a straight line not passing through the center, it also cuts it at right angles; and if it cut it at right angles, it also bisects it. *)

Lemma prop_3_1 : 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.