Library GeoCoq.Highschool.exercises

Require Import GeoCoq.Tarski_dev.Annexes.midpoint_theorems.
Require Import GeoCoq.Highschool.varignon.

Section Exercises.

Context `{TE:Tarski_2D_euclidean}.

Lemma Per_mid_rectangle : A B C I J K,
  A B
  B C
  Per B A C
  Midpoint I B C
  Midpoint J A C
  Midpoint K A B
  Rectangle A J I K.
Proof.
intros.
assert_diffs.
assert_cols.
elim (eq_dec_points A C); intro; apply plg_per_rect.

  treat_equalities.
  assert (HM : M : Tpoint, Midpoint M J I) by (apply midpoint_existence); decompose [ex] HM; repeat split; intuition; x; intuition.

  treat_equalities; intuition.

  assert( Par A B J I Par A C I K Par B C J K
    Cong A K I J Cong B K I J Cong A J I K Cong C J I K Cong B I J K Cong C I J K)
  by (apply triangle_mid_par_cong; intuition).
  spliter.

  elim (Col_dec A B C); intro; assert_diffs.

    apply parallelogram_to_plg; unfold Parallelogram; right; unfold Parallelogram_flat; repeat split.
    ColR.
    ColR.
    assumption.

    assumption.

    right; intro; subst; assert (B = C) by (apply symmetric_point_uniqueness with A K; assumption); contradiction.

    apply pars_par_plg.

      assert (Par_strict A B J I Par_strict A C I K Par_strict B C J K
        Cong A K I J Cong B K I J Cong A J I K Cong C J I K Cong B I J K Cong C I J K)
      by (apply triangle_mid_par_strict_cong; intuition).
      spliter.
      apply par_strict_symmetry; apply par_strict_col_par_strict with C; intuition; apply par_strict_symmetry; apply par_strict_right_comm; assumption; Col.
      Par.
      Col.

      assert (Par_strict A B J I Par_strict A C I K Par_strict B C J K
        Cong A K I J Cong B K I J Cong A J I K Cong C J I K Cong B I J K Cong C I J K)
      by (apply triangle_mid_par_strict_cong; intuition).
      spliter.
      apply par_symmetry; apply par_col_par with B; intuition; apply par_symmetry; apply par_strict_par; assumption.

  left; apply l8_3 with B; try apply l8_2; try apply l8_3 with C; try apply l8_2; try assumption; intuition; Col.

Qed.

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Ltac assert_diffs_by_cases :=
 repeat match goal with
 | A: Tpoint, B: Tpoint |- _not_exist_hyp_comm A B;induction (eq_dec_points A B);[treat_equalities;solve [finish|trivial] |idtac]
end.

Lemma quadrileral_midpoints:
  A B C D I J K L X Y,
  ¬ Col I J K
  Midpoint I A B
  Midpoint J B C
  Midpoint K C D
  Midpoint L A D
  Midpoint X I K
  Midpoint Y J L
  X = Y.
Proof.
intros.
assert_diffs_by_cases.
assert (Parallelogram I J K L)
  by (apply (varignon A B C D I J K L);finish).
assert (Midpoint X J L)
  by (perm_apply (plg_mid_2 I J K L X)).
treat_equalities;trivial.
Qed.

End Exercises.