Require Export GeoCoq.Tarski_dev.Annexes.midpoint_theorems.
Require Export GeoCoq.Highschool.circumcenter.
Require Export GeoCoq.Highschool.orthocenter.
Require Export GeoCoq.Highschool.midpoint_thales.
Require Export GeoCoq.Highschool.concyclic.
Require Export GeoCoq.Highschool.gravityCenter.
Section Euler_line.
Context `{TE:Tarski_euclidean}.
(** <applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar" codebase="http://www.geogebra.org/webstart/4.0/unsigned/" width="796" height="466"> <param name="ggbBase64" 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/> <param name="image" value="http://www.geogebra.org/webstart/loading.gif" /> <param name="boxborder" value="false" /> <param name="centerimage" value="true" /> <param name="java_arguments" value="-Xmx512m -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="4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0, 4.0.19.0" /> <param name="showResetIcon" value="false" /> <param name="showAnimationButton" value="true" /> <param name="enableRightClick" value="false" /> <param name="errorDialogsActive" value="true" /> <param name="enableLabelDrags" value="false" /> <param name="showMenuBar" value="false" /> <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> *)

(* Specifications: *)
Lemma concyclic_not_col_or_eq_aux :
forall A B C D, Concyclic A B C D -> A = B \/ A = C \/ B = C \/ ~ Col A B C.
Lemma concyclic_not_col_or_eq :
forall 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').
Lemma Euler_line_special_case :
forall 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.
Lemma gravity_center_change_triangle:
forall 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'.