(** 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. *)

(* Specifications: *)
Lemma cong2_onc3__eq : forall O P X A B C, A <> B -> A <> C -> B <> C ->
OnCircle A O P -> OnCircle B O P -> OnCircle C O P ->
Cong X A X B -> Cong X A X C ->
X = O.
Lemma onc2_mid_cong_col : forall O P U V M X,
U <> V ->OnCircle U O P -> OnCircle V O P -> Midpoint M U V -> Cong U X V X -> Col O X M.
Lemma cong_onc3_cases : forall O P A X Y,
Cong A X A Y ->
OnCircle A O P -> OnCircle X O P -> OnCircle Y O P ->
X = Y \/ ReflectL X Y O A.
Lemma bet_cong_onc3_cases : forall O P A X Y T,
T <> O -> Bet A O T -> Cong T X T Y ->
OnCircle A O P  -> OnCircle X O P  -> OnCircle Y O P ->
X = Y \/ ReflectL X Y O A.
Lemma prop_7_8 : forall O P A B T X Y , Diam A B O P -> Bet A O T
-> OnCircle X O P -> OnCircle Y O P
-> LeA A O X A O Y -> Le T Y T X.
Lemma Prop_7_8_uniqueness : forall O P A X Y Z T, T <> O -> X <> Y ->
Bet A O T -> Cong T X T Y -> Cong T X T Z ->
OnCircle A O P -> OnCircle X O P -> OnCircle Y O P -> OnCircle Z O P ->
Z = X \/ Z = Y.
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.
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.