Section Exercises.
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> *)
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:
forall 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.