Section Orthocenter.
Context `{TE:Tarski_euclidean}.
(** Orthocenter  # <applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar" codebase="http://jars.geogebra.org/webstart/4.2/unsigned/" width="1669" height="873"> <param name="ggbBase64" value="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" /> <param name="java_arguments" value="-Xmx1024m -Djnlp.packEnabled=true" /> <param name="cache_archive" value="geogebra.jar, geogebra_main.jar, geogebra_gui.jar, geogebra_cas.jar, geogebra_algos.jar, geogebra_export.jar, geogebra_javascript.jar, jlatexmath.jar, jlm_greek.jar, jlm_cyrillic.jar, geogebra_properties.jar" /> <param name="cache_version" value="4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0, 4.2.47.0" /> <param name="showResetIcon" value="false" /> <param name="enableRightClick" value="true" /> <param name="errorDialogsActive" value="true" /> <param name="enableLabelDrags" value="false" /> <param name="showMenuBar" value="true" /> <param name="showToolBar" value="false" /> <param name="showToolBarHelp" value="false" /> <param name="showAlgebraInput" value="false" /> <param name="useBrowserForJS" value="true" /> <param name="allowRescaling" value="true" /> C'est une appliquette Java créée avec GeoGebra ( www.geogebra.org) - Il semble que Java ne soit pas installé sur votre ordinateur, merci d'aller sur www.java.com </applet> #  **)
(* Definition is_orthocenter H A B C := ~ Col A B C /\ exists A1, exists B1, Perp A A1 B C /\ Perp B B1 A C /\ Col H A A1 /\ Col H B B1. *)
Definition is_orthocenter H A B C :=
~ Col A B C /\ Perp A H B C /\ Perp B H A C /\ Perp C H A B.
Proof with finish.
intros.
apply perp_mid_perp_bisect...
apply perp_sym.
apply cop_par_perp__perp with B C...
apply par_col_par with A...
apply perp_col1 with A1...
Qed.
Proof with finish.
intros A A1 B B1 C C1 O HNC HPerp1 HPerp2 HPerp3 HC1 HC2.
assert (HT := HNC).
apply construct_triangle in HT.
destruct HT as [D [E [F HT]]].
spliter.
assert (Midpoint A E F) by (apply diff_not_col_col_par4_mid with B C; finish).
assert (Midpoint B D F) by (apply diff_not_col_col_par4_mid with A C; finish).
assert (Midpoint C D E) by (apply diff_not_col_col_par4_mid with A B; finish).
assert_diffs.
elim (eq_dec_points A O); intro.
treat_equalities; apply col_permutation_4; apply cop_perp2__col with A B...
apply perp_right_comm; apply perp_col1 with B1...
elim (eq_dec_points B O); intro.
treat_equalities; apply col_permutation_4; apply cop_perp2__col with A B...
apply perp_col1 with A1...
elim (eq_dec_points C O); intro.
subst; Col.
assert (Perp_bisect A O E F) by (apply altitude_is_perp_bisect with B C A1; finish).
assert (Perp_bisect B O D F) by (apply altitude_is_perp_bisect with A C B1; finish).
assert (Perp O C D E).
apply circumcenter_intersect with F A B; finish.
apply perp_bisect_sym_1; assumption.
apply perp_bisect_sym_1; assumption.
assert (Perp C1 C D E).
apply perp_sym; apply cop_par_perp__perp with A B...
apply par_symmetry; apply par_col_par_2 with C...
apply col_permutation_2; apply cop_perp2__col with D E; Perp.
apply coplanar_pseudo_trans with A B C; Cop.
apply coplanar_perm_2, col_cop__cop with B1; Col; Cop.
Qed.

Lemma is_orthocenter_cases :
forall A B C G,
is_orthocenter G A B C \/
is_orthocenter G A C B \/
is_orthocenter G B A C \/
is_orthocenter G B C A \/
is_orthocenter G C A B \/
is_orthocenter G C B A ->
is_orthocenter G A B C.