(** another statement for nine point circle theorem *)

(* Specifications: *)
Lemma nine_point_1 : forall A B C A' B' C' M,
is_midpoint A' B C ->
is_midpoint B' A C ->
is_midpoint C' A B ->
on_foot M A B C ->
co_circle M A' B' C'.
Lemma l_6_265 : forall A B C M F H,
is_midpoint M B C ->
on_perp_d F A B 1 ->
on_perp_d H A C (- (1)) ->
Py H F H = (2+2) * Py M A M.
Lemma l3_51: forall A B C D J M N,
on_foot D A B C ->
on_line J A D ->
inter_ll M A B C J ->
inter_ll N A C B J ->
eq_angle M D A A D N.
Lemma l6_17 : forall A B R P Q S C F,
on_line P B R ->
on_line Q A R ->
inter_ll S A P B Q ->
inter_ll C R S A B ->
inter_ll F P Q A B ->
parallel A C B C ->
parallel A F B F ->
B<>C -> B<>F ->
A**C / B**C = - A**F / B**F.
Lemma l_6_272 : forall A B C M E P G Q L,
is_midpoint M B C ->
on_perp_d E A B 1 ->
is_midpoint P B E ->
on_perp_d G A C (0-1) ->
is_midpoint Q C G ->
is_midpoint L E G ->
eq_distance P M L Q.
Lemma l6_84 : forall  A B C D E F G,
on_foot D C A B ->
on_foot E B A C ->
is_midpoint F B C ->
is_midpoint G D E ->
perp G F D E.
Lemma l_3_45 : forall A B C E F G A1 B1 C1 r,
is_midpoint E A C ->
on_perp_d B1 E A r ->
is_midpoint F B C ->
on_perp_d A1 F C r ->
is_midpoint G A B ->
on_perp_d C1 G B (0-r) ->
parallel A1 C1 C B1.
Lemma l6_46 : forall A B C F N K J,
is_midpoint F A B ->
on_line N C F ->
on_foot K N A C ->
on_foot J N B C ->
Py N K N * Py A C A = Py N J N * Py B C B.
Lemma l6_44 : forall A B C G P Q I J,
is_midpoint J A C ->
is_midpoint I B C ->
inter_ll G J B I A ->
on_inter_line_parallel P G A B B C ->
on_inter_line_parallel Q G A C B C ->
G<>P ->
G<>Q ->
(2+2)*S4 P Q C B = (2+2+1)*S A Q P.
Theorem th6_259 :
forall A B C D E F G H R L : Point,
on_parallel_d E A B D 1 ->
on_parallel_d G A C F 1 ->
inter_ll H D E F G ->
inter_ll R B C H A ->
on_parallel_d L B H A 1 ->
S F A B <> 0 ->
S A C F + S B A D = S C B L.