Set Implicit Arguments.
Unset Strict Implicit.
Require Export point_cocyclicite.
unfold colineaire in |- *; intros.
apply regulier with (a := double (cons u v)) (c := double (cons u v)); auto.
apply transitive with (double (plus (cons u v) (cons v u))); auto.
apply transitive with (double zero); auto.
apply transitive with (plus zero zero); auto.
apply compatible; auto.
Qed.
Hint Resolve colineaire_sym.
unfold colineaire in |- *; intros.
apply
transitive with (double (plus (cons u' u) (plus (cons u v) (cons v v'))));
auto.
apply
transitive
with (plus (double (cons u' u)) (double (plus (cons u v) (cons v v'))));
auto.
apply
transitive
with
(plus (double (cons u' u))
(plus (double (cons u v)) (double (cons v v'))));
auto.
apply compatible; auto.
apply
transitive with (plus zero (plus (double (cons u v)) (double (cons v v'))));
auto.
apply compatible; auto.
cut (colineaire u' u); intros; auto.
apply transitive with (plus (double (cons u v)) (double (cons v v'))); auto.
apply transitive with (plus (double (cons u v)) zero); auto.
apply compatible; auto.
apply transitive with (plus zero (double (cons u v))); auto.
Qed.
Hint Resolve colineaire_modulo_pi.
unfold orthogonal in |- *; intros.
apply transitive with (double (plus (cons u v) (cons v (opp v)))); auto.
apply transitive with (plus (double (cons u v)) (double (cons v (opp v))));
auto.
apply transitive with (plus pi (double pi)); auto.
apply compatible; auto.
apply transitive with (plus pi zero); auto.
apply compatible; auto.
apply transitive with (plus zero pi); auto.
Qed.
Hint Resolve orthogonal_opp.
unfold colineaire, orthogonal in |- *; intros.
apply transitive with (double (plus (cons u v) (cons v w))); auto.
apply transitive with (plus (double (cons u v)) (double (cons v w))); auto.
apply transitive with (plus pi zero); auto.
apply compatible; auto.
apply transitive with (plus zero pi); auto.
Qed.
intros A B C M P Q S H H0 H1 H2 H3 H4; try assumption.
cut
(R (double (cons (vec P Q) (vec P M))) (double (cons (vec C A) (vec C M))));
intros.
cut
(R (double (cons (vec P S) (vec P M))) (double (cons (vec B A) (vec B M))));
intros.
apply
transitive
with (double (plus (cons (vec P Q) (vec P M)) (cons (vec P M) (vec P S))));
auto.
apply
transitive
with
(plus (double (cons (vec P Q) (vec P M)))
(double (cons (vec P M) (vec P S)))); auto.
apply compatible; auto.
apply transitive with (double (cons (vec B S) (vec B M))).
apply symetrique; apply (deux_rectangles (A:=M) (B:=B) (C:=S) (D:=P)); auto.
apply colineaire_modulo_pi; auto.
unfold colineaire in |- *.
apply transitive with (double zero); auto.
apply transitive with (double (cons (vec C Q) (vec C M))).
apply symetrique; apply (deux_rectangles (A:=M) (B:=C) (C:=Q) (D:=P)); auto.
apply orthogonal_colineaire with (vec P B); auto.
apply colineaire_modulo_pi; auto.
unfold colineaire in |- *.
apply transitive with (double zero); auto.
Qed.

Theorem droite_Simson :
forall A B C M P Q T : PO,
~ colineaire (vec C A) (vec C M) ->
colineaire (vec C A) (vec C Q) ->
colineaire (vec P C) (vec P B) ->
colineaire (vec B A) (vec B T) ->
orthogonal (vec T M) (vec T B) ->
orthogonal (vec P M) (vec P B) ->
orthogonal (vec Q M) (vec Q C) ->
(colineaire (vec P Q) (vec P T) <-> sont_cocycliques C A M B).