Library GeoCoq.Highschool.Euler_line

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Lemma concyclic_not_col_or_eq_aux :
   A B C D, Concyclic A B C D A = B A = C B = C ¬ Col A B C.
Proof.
intros A B C D HC.
elim (eq_dec_points A B); intro HAB; Col.
elim (eq_dec_points A C); intro HAC; Col.
elim (eq_dec_points B C); intro HBC; Col5.
right; right; right; intro HCol.
destruct HC as [HCop [O [HCong1 [HCong2 HCong3]]]].
assert (H := midpoint_existence A B); destruct H as [M1 HMid1].
assert (HOM1 : O M1).
  {
  intro; treat_equalities.
  assert (H := l7_20 O A C); elim H; clear H; try intro H; Cong;
  try (apply HBC; apply symmetric_point_uniqueness with A O; Col);
  assert_cols; ColR.
  }
assert (H := midpoint_existence A C); destruct H as [M2 HMid2].
assert (HOM2 : O M2).
  {
  intro; treat_equalities.
  assert (H := l7_20 O A B); elim H; clear H; try intro H; Cong;
  try (apply HBC; apply symmetric_point_uniqueness with A O; Col);
  assert_cols; ColR.
  }
assert (HM1M2 : M1 M2) by (intro; treat_equalities; Col).
assert (HPerp1 : Perp_bisect O M1 A B)
  by (apply cong_perp_bisect; unfold Midpoint in *; spliter; Cong).
assert (HPerp2 : Perp_bisect O M2 A C)
  by (apply cong_perp_bisect; unfold Midpoint in *; spliter; Cong).
assert (HOM1M2 : ¬ Col O M1 M2).
  {
  intro HOM1M2; assert (H := l7_20 O A B); elim H; clear H; try intro H; Cong;
  try (apply HOM1; apply l7_17 with A B; Col); assert_diffs; assert_cols; ColR.
  }
assert (HPar_strict : Par_strict O M1 O M2).
  {
  apply par_not_col_strict with M2; Col.
  apply l12_9 with A B; Col;
  try (apply perp_col0 with A C; Col; perm_apply perp_bisect_perp);
  apply perp_bisect_perp; Col.
  }
assert (H := not_par_strict_id O M1 M2); Col.
Qed.

Lemma concyclic_not_col_or_eq :
   A B C A', Concyclic A B C A'
  A'=C A'=B A=B A=C A=A' (¬ Col A B A' ¬ Col A C A').
Proof.
intros A B C A' H.
assert (H' := H); apply concyclic_perm_1 in H; apply concyclic_perm_3 in H'.
apply concyclic_not_col_or_eq_aux in H; apply concyclic_not_col_or_eq_aux in H'.
elim (eq_dec_points A' C); intro; try tauto.
elim (eq_dec_points A' B); intro; try tauto.
elim (eq_dec_points A B); intro; try tauto.
elim (eq_dec_points A C); intro; try tauto.
elim (eq_dec_points A A'); intro; try tauto.
do 3 (elim H; clear H; intro H; try tauto); Col.
do 3 (elim H'; clear H'; intro H'; try tauto); Col.
Qed.

Lemma Euler_line_special_case :
   A B C G H O,
  Per A B C
  is_gravity_center G A B C
  is_orthocenter H A B C
  is_circumcenter O A B C
  Col G H O.
Proof.
intros.
assert (H=B).
  apply orthocenter_per with A C;finish.
subst.
assert (Midpoint O A C).
 apply circumcenter_per with B;finish.
 unfold is_orthocenter in *;spliter;assert_diffs;finish.
 unfold is_orthocenter in *;spliter;assert_diffs;finish.
assert (is_gravity_center G A C B).
 apply is_gravity_center_perm in H1;intuition.
perm_apply (is_gravity_center_col A C B G O).
Qed.

Lemma gravity_center_change_triangle:
  A B C G I B' C',
 is_gravity_center G A B C
 Midpoint I B C
 Midpoint I B' C'
 ¬ Col A B' C'
 is_gravity_center G A B' C'.
Proof.
intros.
Name G' the midpoint of A and G.
assert (Midpoint G I G')
  by (apply (is_gravity_center_third A B C G G' I);finish).
apply (is_gravity_center_third_reci A B' C' G I G');finish.
Qed.

Hint Resolve
     is_gravity_center_perm_1
     is_gravity_center_perm_2
     is_gravity_center_perm_3
     is_gravity_center_perm_4
     is_gravity_center_perm_5 : gravitycenter.

Hint Resolve
     is_orthocenter_perm_1
     is_orthocenter_perm_2
     is_orthocenter_perm_3
     is_orthocenter_perm_4
     is_orthocenter_perm_5 : Orthocenter.

Hint Resolve
     is_circumcenter_perm_1
     is_circumcenter_perm_2
     is_circumcenter_perm_3
     is_circumcenter_perm_4
     is_circumcenter_perm_5 : Circumcenter.

Lemma Euler_line :
  A B C G H O,
  ¬ Col A B C
  is_gravity_center G A B C
  is_orthocenter H A B C
  is_circumcenter O A B C
  Col G H O.
Proof.
intros.
elim (Cong_dec A B A C); intro.

Name A' the midpoint of B and C.
Name B' the midpoint of A and C.
Name C' the midpoint of A and B.

assert (Perp_bisect A A' B C)
  by (apply cong_perp_bisect; assert_diffs; unfold Midpoint in *; spliter;
      Cong; intro; treat_equalities; assert_cols; Col).

assert (Col G A' A)
  by (apply is_gravity_center_perm in H1; apply is_gravity_center_col with B C; spliter; Col).

unfold is_orthocenter in *; spliter.

elim (eq_dec_points O G); intro; treat_equalities; Col;
elim (eq_dec_points O H); intro; treat_equalities; Col.

elim (eq_dec_points O A'); intro; treat_equalities.

assert (Col A H O) by (apply perp_perp_col with B C; Col; apply perp_bisect_perp; Col).
apply col_permutation_1; apply perp_perp_col with B C.
apply perp_sym; apply perp_col0 with A O; try apply perp_bisect_perp; Col.
apply perp_sym; apply perp_col0 with A H; Col.

assert (Col A A' H) by (apply perp_perp_col with B C; Col; apply perp_bisect_perp; Col).
assert (Perp_bisect O A' B C) by (apply circumcenter_perp with A; assert_diffs; Col).
assert (Col A' A O)
  by (apply perp_perp_col with B C; apply perp_left_comm; apply perp_bisect_perp; Col).
show_distinct A A'; assert_cols; Col; ColR.

Name A' the symmetric of A wrt O.

assert_diffs.

assert (Concyclic A B C A').
 unfold Concyclic.
 split.
 apply all_coplanar.
  O.
 apply circumcenter_cong in H3.
 spliter.
 assert_congs_perm.
 spliter;repeat (split;finish).

assert (T:=concyclic_not_col_or_eq A B C A' H5).
decompose [or] T;clear T;try contradiction.
 - subst.
   assert (Per A B C).
    apply midpoint_thales with O;finish.
    unfold is_circumcenter in *;spliter;finish.
   apply (Euler_line_special_case A B C G H O);finish.
 - subst.
   assert (Per A C B).
    apply midpoint_thales with O;finish.
    unfold is_circumcenter in *;spliter.
    apply cong_transitivity with O B;finish.

   apply (Euler_line_special_case A C B G H O);finish.
   try (apply is_gravity_center_perm_1; assumption).
   auto with Orthocenter.
   auto with Circumcenter.

 - unfold is_circumcenter in *;spliter.
   treat_equalities.
   intuition.
 - spliter.

assert_diffs.

assert (Per A B A').
 apply midpoint_thales with O;finish.
 unfold is_circumcenter in *;spliter;finish.

assert (Perp C H A B)
 by (unfold is_orthocenter in *;spliter;finish).

assert (Perp A' B B A)
 by (apply per_perp;finish).

assert (Par C H A' B)
 by (apply l12_9 with A B;finish).

assert (Perp B H A C)
 by (unfold is_orthocenter in *;spliter;finish).

assert (Per A C A').
 {
 apply midpoint_thales with O;finish.
 unfold is_circumcenter in *;spliter;finish.
 apply cong_transitivity with B O;finish.
 }

assert (Perp A' C C A) by (apply per_perp;finish).

assert (Par B H C A')
 by (apply l12_9 with A C;finish).

induction (Col_dec B H C).
 × assert (H=B H=C) by (apply (orthocenter_col A B C H);finish).
   induction H26.
   + subst H.
     assert (Midpoint O A C) by (apply (circumcenter_per) with B;finish).
     assert (Col G O B).
        apply (is_gravity_center_col A C B G O).
        apply is_gravity_center_perm in H1;intuition idtac.
        assumption.
     Col.
   + subst H;assert_diffs; intuition.
 × assert (Parallelogram B H C A')
     by (apply par_2_plg;finish).

   assert (T: I : Tpoint, Midpoint I B C Midpoint I H A')
     by (apply plg_mid;finish).

   destruct T as [I [HI1 HI2]].

   elim (Per_dec B A C); intro.

   apply Euler_line_special_case with B A C;
   try apply is_gravity_center_cases; auto;
   try apply is_orthocenter_cases; auto;
   try apply is_circumcenter_cases; auto.

   assert (is_gravity_center G A H A').
     {
     apply gravity_center_change_triangle with B C I;finish.
     show_distinct A' H; treat_equalities.
     apply plg_par in H26; spliter; assert_diffs; Col.
     apply H25; apply par_id_5; Par.
     intro.
     Name A'' the midpoint of B and C.
     show_distinct A'' O; treat_equalities.
     apply H27; apply perp_per_1; assert_diffs; Perp.
     assert (Perp_bisect O A'' B C) by (apply circumcenter_perp with A; Col).
     elim (eq_dec_points A A''); intro; treat_equalities.
     apply perp_bisect_cong_2 in H32; apply H4; Cong.
     assert (Perp_bisect A'' A B C).
     apply perp_mid_perp_bisect; Col.
     apply perp_sym; apply perp_col0 with O A''; Col;
     try (apply perp_bisect_perp; assumption); assert_cols; try ColR.
     apply perp_bisect_cong_2 in H34; apply H4; Cong.
     }

   assert (is_gravity_center G A A' H)
     by (apply is_gravity_center_cases;auto).

   perm_apply (is_gravity_center_col A A' H G O).
Qed.

End Euler_line.