(** Prop 7   **)

(* Specifications: *)
Lemma onc_sym__onc : forall O P A B X Y,
Bet O A B -> OnCircle A O P -> OnCircle B O P -> OnCircle X O P -> ReflectL X Y A B -> OnCircle Y O P.
Lemma mid_onc__diam : forall O P A B, OnCircle A O P -> Midpoint O A B -> Diam A B O P.
Lemma chord_le_diam : forall O P A B U V,
Diam A B O P -> OnCircle U O P -> OnCircle V O P -> Le U V A B.
Lemma chord_lt_diam : forall O P A B U V,
~ Col O U V -> Diam A B O P -> OnCircle U O P -> OnCircle V O P ->
Lt U V A B.
Lemma inc2_le_diam: forall O P A B U V, Diam A B O P -> InCircle U O P -> InCircle V O P -> Le U V A B.
Lemma onc_col_diam__eq : forall O P A B X, Diam A B O P -> OnCircle X O P -> Col A B X -> X = A \/ X = B.
Lemma onc2_out__eq : forall O P A B, OnCircle A O P -> OnCircle B O P -> Out O A B -> A = B.
Lemma bet_onc_le_a : forall O P A B T X, Diam A B O P -> Bet B O T -> OnCircle X O P -> Le T A T X.
Lemma bet_onc_lt_a : forall O P A B T X,
Diam A B O P -> O <> P -> O <> T -> X <> A -> Bet B O T  -> OnCircle X O P ->
Lt T A T X.
Lemma bet_onc_le_b : forall O P A B T X,
Diam A B O P -> Bet A O T -> OnCircle X O P ->
Le T X T A.
Lemma bet_onc_lt_b : forall O P A B T X,
Diam A B O P -> T <> O -> X <> A -> Bet A O T -> OnCircle X O P ->
Lt T X T A.
Lemma incs2_lt_diam : forall O P A B U V, Diam A B O P -> InCircleS U O P -> InCircleS V O P -> Lt U V A B.
Lemma incs_onc_diam__lt : forall O P A B U V, Diam A B O P -> InCircleS U O P -> OnCircle V O P -> Lt U V A B.
Lemma diam_cong_incs__outcs : forall O P A B U V, Diam A B O P -> Cong A B U V -> InCircleS U O P -> OutCircleS V O P.
Lemma diam_uniqueness : forall O P A B X, Diam A B O P -> Cong A X A B -> OnCircle X O P -> X = B.
Lemma onc3__ncol : forall O P A B C,
A <> B -> A <> C -> B <> C ->
OnCircle A O P -> OnCircle B O P -> OnCircle C O P ->
~ Col A B C.
Lemma diam_exists : forall O P T, exists A, exists B, Diam A B O P /\ Col A B T.
Lemma chord_intersection : forall O P A B X Y,
OnCircle A O P -> OnCircle B O P -> OnCircle X O P -> OnCircle Y O P -> TS A B X Y ->
TS X Y A B.
Lemma ray_cut_chord : forall O P A B X Y,
Diam A B O P -> OnCircle X O P -> OnCircle Y O P -> TS A B X Y -> OS X Y O A ->
TS X Y O B.
Lemma center_col__diam : forall O P A B,
A <> B -> Col O A B -> OnCircle A O P -> OnCircle B O P ->
Diam A B O P.
Lemma diam__midpoint: forall O P A B, Diam A B O P -> Midpoint O A B.
Lemma diam_sym : forall O P A B, Diam A B O P -> Diam B A O P.
Lemma diam_end_uniqueness : forall O P A B C, Diam A B O P -> Diam A C O P -> B = C.
Lemma center_onc2_mid__ncol : forall O P A B M ,
O <> P -> ~ Col O A B ->
OnCircle A O P -> OnCircle B O P ->
Midpoint M A B  -> ~ Col A O M.
Lemma bet_chord__diam_or_ncol : forall O P A B T,
A <> B -> T <> A -> T <> B -> OnCircle A O P -> OnCircle B O P -> Bet A T B ->
Diam A B O P \/ ~Col O A T /\ ~Col O B T.
Lemma mid_chord__diam_or_ncol : forall O P A B T,
A <> B -> OnCircle A O P -> OnCircle B O P ->
Midpoint T A B ->
Diam A B O P \/ ~Col O A T /\ ~Col O B T.
Lemma cop_mid_onc2_perp__col : forall O P A B X Y, A <> B -> OnCircle A O P -> OnCircle B O P ->
Midpoint X A B -> Perp X Y A B -> Coplanar O A B Y -> Col X Y O.
Lemma cong2_cop2_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 -> Coplanar A B C O ->
Cong X A X B -> Cong X A X C -> Coplanar A B C X ->
X = O.
Lemma tree_points_onc_cop : forall O P, O <> P -> exists A B C,
A <> B /\ A <> C /\ B <> C /\ OnCircle A O P /\ OnCircle B O P /\ OnCircle C O P /\ Coplanar A B C O.
Lemma tree_points_onc_cop2 : forall O P Q, O <> P -> exists A B C,
A <> B /\ A <> C /\ B <> C /\ OnCircle A O P /\ OnCircle B O P /\ OnCircle C O P /\
Coplanar A B C O /\ Coplanar A B C Q.
Lemma tree_points_onc : forall O P, O <> P -> exists A B C,
A <> B /\ A <> C /\ B <> C /\ OnCircle A O P /\ OnCircle B O P /\ OnCircle C O P.
Lemma bet_cop_onc2__ex_onc_os_out : forall O P A B C I,
A <> I -> B <> I -> ~ Col A B C -> ~ Col A B O ->
OnCircle A O P -> OnCircle B O P -> Bet A I B -> Coplanar A B C O ->
exists C1, Out C1 O I /\ OnCircle C1 O P /\ OS A B C C1.