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" value="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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> *)

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'.