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.
unfold colineaire 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 zero zero); auto.
apply compatible; auto.
Qed.
unfold orthogonal 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 pi (plus (double (cons u' v')) pi)); auto.
apply compatible; auto.
apply compatible; auto.
apply regulier with (a := double (cons v v')) (c := pi); auto.
apply transitive with (double (plus (cons v v') (cons v' v))); auto.
apply transitive with (double (cons v v)); auto.
apply transitive with (double zero); auto.
apply transitive with zero; auto.
apply transitive with (plus (plus (double (cons u' v')) pi) pi); auto.
apply transitive with (plus (double (cons u' v')) (plus pi pi)); 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 double_orthogonal.
intros u v H; unfold orthogonal, double in |- *.
apply transitive with (plus (cons u v) (cons v (opp u))); auto.
apply compatible; auto.
apply transitive with (cons u (opp u)); auto.
Qed.
unfold orthogonal, double in |- *; intros u v H.
apply transitive with (plus (cons v u) pi); auto.
apply regulier with (a := cons u v) (c := cons u v); auto.
apply transitive with pi; auto.
apply transitive with (plus (plus (cons u v) (cons v u)) pi); auto.
apply transitive with (plus zero pi); auto.
apply compatible; auto.
Qed.
Hint Resolve critere_orthogonal critere_orthogonal_reciproque.
Definition bissectrice (I A B C : PO) :=
R (cons (vec A B) (vec A I)) (cons (vec A I) (vec A C)).
unfold bissectrice, double in |- *; intros.
apply
transitive with (plus (cons (vec A B) (vec A I)) (cons (vec A I) (vec A C)));
auto.
apply compatible; auto.
Qed.
Hint Resolve bissectrice_double.
unfold colineaire in |- *; intros.
apply
transitive
with (double (plus (cons (vec A I) (vec A B)) (cons (vec A B) (vec A J))));
auto.
apply
transitive
with
(plus (double (cons (vec A I) (vec A B)))
(double (cons (vec A B) (vec A J)))); auto.
apply
transitive with (plus (cons (vec A C) (vec A B)) (cons (vec A B) (vec A C)));
auto.
apply compatible; unfold double in |- *.
apply
transitive with (plus (cons (vec A C) (vec A I)) (cons (vec A I) (vec A B)));
auto.
apply compatible; auto.
apply
transitive with (plus (cons (vec A B) (vec A J)) (cons (vec A J) (vec A C)));
auto.
apply compatible; auto.
Qed.
Hint Resolve bissectrice_unicite.
unfold colineaire in |- *; intros.
apply
transitive
with (double (plus (cons (vec A I) (vec A B)) (cons (vec A B) u)));
auto.
apply
transitive
with (plus (double (cons (vec A I) (vec A B))) (double (cons (vec A B) u)));
auto.
apply
transitive with (plus (cons (vec A C) (vec A B)) (cons (vec A B) (vec A C)));
auto.
apply compatible; unfold double in |- *.
apply
transitive with (plus (cons (vec A C) (vec A I)) (cons (vec A I) (vec A B)));
auto.
apply compatible; auto.
apply transitive with (plus (cons (vec A B) u) (cons u (vec A C))); auto.
apply compatible; auto.
Qed.
Hint Resolve bissectrice_direction.
unfold isocele, orthogonal, bissectrice in |- *; intros I A B C H H0.
apply
transitive
with
(double
(plus (cons (vec A I) (vec A C))
(plus (cons (vec A C) (vec C A)) (cons (vec C A) (vec B C)))));
auto.
apply
transitive
with
(plus (double (cons (vec A I) (vec A C)))
(double (plus (cons (vec A C) (vec C A)) (cons (vec C A) (vec B C)))));
auto.
apply
transitive
with
(plus (cons (vec A B) (vec A C))
(plus (double (cons (vec A C) (vec C A)))
(double (cons (vec C A) (vec B C))))); auto.
apply compatible; auto.
apply
transitive
with
(plus (cons (vec A B) (vec A C))
(plus (double pi) (double (cons (vec C A) (vec B C)))));
auto.
apply compatible; auto.
apply compatible; auto.
apply
transitive
with
(plus (cons (vec A B) (vec A C))
(plus zero (double (cons (vec C A) (vec B C)))));
auto.
apply compatible; auto.
apply compatible; auto.
apply
transitive
with (plus (cons (vec A B) (vec A C)) (double (cons (vec B C) (vec B A))));
auto.
apply compatible; auto.
apply transitive with (double (cons (vec C A) (vec C B))); auto.
apply transitive with (double (cons (vec C A) (vec B C))); auto.
apply
transitive
with (double (plus (cons (vec C A) (vec C B)) (cons (vec C B) (vec B C))));
auto.
apply
transitive
with
(plus (double (cons (vec C A) (vec C B)))
(double (cons (vec C B) (vec B C)))); auto.
apply transitive with (plus (double (cons (vec C A) (vec C B))) zero); auto.
apply compatible; auto.
apply transitive with (double pi); auto.
apply transitive with (plus zero (double (cons (vec C A) (vec C B)))); auto.
apply triangle_isocele; auto.
Qed.
intros u v H; try assumption.
apply regulier with (a := cons u v) (c := cons u v); auto.
apply transitive with (plus (cons (opp v) u) (cons u v)); auto.
apply transitive with (cons (opp v) v); auto.
apply transitive with pi; auto.
Qed.
Hint Resolve orthogonal_bissectrice.
unfold isocele, bissectrice in |- *; intros I A B C H H0.
apply reflexion with (vec A I); auto.
apply transitive with (cons (opp (vec B C)) (vec A I)); auto.
apply vR_R_compatible; auto.
apply v_sym; apply opp_vect.
apply
transitive with (plus (cons (vec C A) (vec A C)) (cons (vec A C) (vec A I)));
auto.
apply
transitive with (plus (cons (vec A I) (vec A B)) (cons (vec A B) (vec B A)));
auto.
apply
transitive with (plus (cons (vec A B) (vec B A)) (cons (vec A I) (vec A B)));
auto.
apply compatible; auto.
apply transitive with pi; auto.
Qed.
unfold bissectrice, isocele in |- *; intros.
generalize (critere_orthogonal_reciproque (u:=vec B C) (v:=vec A I));
intros H2; try exact H2.
apply
transitive with (plus (cons (vec A B) (vec C B)) (cons (vec C B) (vec A I)));
auto.
apply
transitive
with (plus (cons (vec B C) (vec A C)) (cons (opp (vec B C)) (vec A I)));
auto.
apply compatible; auto.
apply transitive with (cons (opp (vec B A)) (opp (vec B C))); auto.
apply vR_R_compatible; auto.
apply v_sym; apply opp_vect.
apply v_sym; apply opp_vect.
apply transitive with (cons (opp (vec C B)) (opp (vec C A))); auto.
apply transitive with (cons (vec C B) (vec C A)); auto.
apply transitive with (cons (vec B A) (vec B C)); auto.
apply vR_R_compatible; auto.
apply opp_vect.
apply opp_vect.
apply vR_R_compatible; auto.
apply v_sym; apply opp_vect.
apply
transitive with (plus (cons (vec B C) (vec A C)) (cons (vec A I) (vec B C)));
auto.
apply compatible; auto.
apply
transitive with (plus (cons (vec A I) (vec B C)) (cons (vec B C) (vec A C)));
auto.
Qed.
unfold isocele, bissectrice in |- *; intros I A B C D H H0 H1.
cut (R (cons (vec B C) (vec A I)) (cons (vec A I) (vec C B))); intros.
cut (R (cons (vec B D) (vec B C)) (cons (vec C B) (vec C D))); intros; auto.
apply
transitive with (plus (cons (vec D B) (vec B D)) (cons (vec B D) (vec A I)));
auto.
apply
transitive with (plus (cons (vec A I) (vec I A)) (cons (vec I A) (vec D C)));
auto.
apply compatible; auto.
apply transitive with pi; auto.
apply
transitive with (plus (cons (vec B D) (vec B C)) (cons (vec B C) (vec A I)));
auto.
apply
transitive with (plus (cons (vec C B) (vec C D)) (cons (vec B C) (vec A I)));
auto.
apply compatible; auto.
apply
transitive with (plus (cons (vec C B) (vec C D)) (cons (vec A I) (vec C B)));
auto.
apply compatible; auto.
apply
transitive with (plus (cons (vec A I) (vec C B)) (cons (vec C B) (vec C D)));
auto.
apply transitive with (cons (vec A I) (vec C D)); auto.
apply transitive with (cons (opp (vec I A)) (opp (vec D C))); auto.
apply vR_R_compatible; auto.
apply v_sym; apply opp_vect.
apply v_sym; apply opp_vect.
cut (orthogonal (vec A I) (vec B C)); intros; auto.
apply transitive with (cons (vec A I) (opp (vec B C))); auto.
apply vR_R_compatible; auto.
apply opp_vect.
apply isocele_bissectrice_hauteur; auto.
Qed.
unfold colineaire in |- *; intros u v w t H H0.
apply transitive with (plus (cons v u) (cons u t)); auto.
apply transitive with (plus (cons u w) (cons u t)); auto.
apply compatible; auto.
apply regulier with (a := cons u t) (c := cons u t); auto.
apply transitive with (cons u w); auto.
apply transitive with (plus (cons u t) (plus (cons u t) (cons u w))); auto.
apply compatible; auto.
apply transitive with (plus (plus (cons u t) (cons u t)) (cons u w)); auto.
apply transitive with (plus zero (cons u w)); auto.
apply compatible; auto.
Qed.
Hint Resolve bissectrice_droite.
Definition milieu (I B C : PO) :=
colineaire (vec B I) (vec B C) /\
(forall A : PO, isocele A B C -> bissectrice I A B C).
unfold milieu in |- *; intros.
elim H0; intros H1 H2; clear H0; try exact H2.
apply H2; auto.
Qed.
Axiom
point_aligne :
forall A B C : PO,
colineaire (vec A B) (vec C B) -> colineaire (vec A B) (vec A C).
intros A B C D H H0; try assumption.
elim existence_milieu with (B := B) (C := C); intros I H1; try exact H1.
cut (bissectrice I A B C /\ bissectrice I D B C); intros.
elim H2; intros H3 H4; clear H2; try exact H3.
cut (colineaire (vec A I) (vec D I)); intros.
unfold bissectrice in |- *.
split; [ idtac | try assumption ].
apply bissectrice_droite with (vec A I); auto.
apply point_aligne; auto.
apply bissectrice_droite with (vec D I); auto.
apply point_aligne; auto.
apply bissectrice_direction with (1 := H3); auto.
apply bissectrice_deux_isoceles; auto.
split; [ idtac | try assumption ].
apply milieu_isocele; auto.
apply milieu_isocele; auto.
Qed.
unfold circonscrit in |- *; intros A B C P Q T O O' I H H0 H1 H2 H3 H4.
apply reciproque_cocyclique; auto.
apply symetrique.
apply
transitive
with (double (plus (cons (vec I B) (vec I A)) (cons (vec I A) (vec I C))));
auto.
apply
transitive
with (double (plus (cons (vec T B) (vec T A)) (cons (vec Q A) (vec Q C))));
auto.
apply
regulier
with
(a := double (cons (vec P C) (vec P B)))
(c := double (cons (vec P C) (vec P B))); auto.
apply
transitive
with (double (plus (cons (vec P C) (vec P B)) (cons (vec P B) (vec P C))));
auto.
apply transitive with (double (cons (vec P C) (vec P C))); auto.
apply transitive with (double zero); auto.
apply
transitive
with
(double
(plus (cons (vec P C) (vec P B))
(plus (cons (vec T B) (vec T A)) (cons (vec Q A) (vec Q C)))));
auto.
apply transitive with (double pi); auto.
apply transitive with zero; auto.
apply
transitive
with
(plus (double (cons (vec T B) (vec T A)))
(double (cons (vec Q A) (vec Q C)))); auto.
apply
transitive
with
(plus (double (cons (vec I B) (vec I A)))
(double (cons (vec I A) (vec I C)))); auto.
apply compatible.
apply cocyclique with O.
elim H0; intros H5 H6; clear H0; auto.
elim H0; intros H5 H6; elim H6; intros H7 H8; clear H6 H0; auto.
elim H0; intros H5 H6; elim H6; intros H7 H8; clear H6 H0; auto.
elim H; intros H5 H6; elim H6; intros H7 H8; clear H6 H; auto.
elim H; intros H5 H6; elim H6; intros H7 H8; clear H6 H; auto.
apply cocyclique with O'.
elim H2; intros H5 H6; clear H2; try exact H5.
elim H2; intros H5 H6; elim H6; intros H7 H8; clear H6 H2; auto.
elim H2; intros H5 H6; elim H6; intros H7 H8; clear H6 H2; auto.
elim H1; intros H5 H6; elim H6; intros H7 H8; clear H6 H1; auto.
elim H1; intros H5 H6; elim H6; intros H7 H8; clear H6 H1; auto.
Qed.

Lemma circonscrit_3centres :
forall A B C O1 O2 O3 I : PO,
circonscrit I A B O1 ->
circonscrit I A C O2 ->
circonscrit I B C O3 ->
R (double (cons (vec O1 O2) (vec O1 O3)))
(double (cons (vec I A) (vec I B))) /\
R (double (cons (vec O1 O3) (vec O2 O3)))
(double (cons (vec I B) (vec I C))) /\
R (double (cons (vec O1 O2) (vec O2 O3)))
(double (cons (vec I A) (vec I C))).