Section GravityCenter.
Context `{TE:Tarski_euclidean}.
(** Center of gravity  # <applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar" codebase="http://jars.geogebra.org/webstart/4.2/unsigned/" width="917" height="865"> <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="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> #  **)
Proof with finish.
intros.
assert_bets.
elim (inner_pasch A B C J I)...
intro G;intros.
exists G.
spliter;assert_cols;split...
Qed.
Proof with finish.
intros.
elim (intersection_two_medians_exist A B C I J H H0 H1); intros G HG; spliter.
exists G.
unfold unique.
assert_all.
repeat split...
intros G' HG'; spliter.
apply l6_21 with A I B J...
intro; search_contradiction.
show_distinct' B J...
Qed.
Definition is_gravity_center G A B C :=
~ Col A B C /\
exists I, exists J, Midpoint I B C /\ Midpoint J A C /\ Col G A I /\ Col G B J.

Lemma is_gravity_center_exist_unique : forall A B C,
~ Col A B C ->
exists! G, is_gravity_center G A B C.