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="UEsDBBQACAAIAIFrzEIAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT/LP88zLLNHQVKiuBQBQSwcI1je9uRkAAAAXAAAAUEsDBBQACAAIAIFrzEIAAAAAAAAAAAAAAAAMAAAAZ2VvZ2VicmEueG1s1Vrbcts4En3OfAWKzzGFKy8pKVNOUpN1ypmk1tmtrX0DSVjCmCK5JOXLVD5+GwBJkZKdjGxnV6EjgwAbaPTpxummnPmvt+scXau60WWx8IiPPaSKtMx0sVx4m/byJPJ+ff3LfKnKpUpqiS7Lei3bhcd96m3nQc/noZmss4UXciaCTCYnPFXpCZdKniSChCchkbHEUjAaBx5Ct41+VZS/y7VqKpmqi3Sl1vK8TGVr11y1bfVqNru5ufF77X5ZL2fLZeLfNpmHYOdFs/C6m1ew3GTSDbPiFGMy+9fHc7f8iS6aVhap8pCxaqNf//JifqOLrLxBNzprVwsvZmDGSunlCsyM49hDMyNUga2VSlt9rRqYOupam9t15VkxWZjnL9wdygdzPJTpa52peuFhnxKBBQsDxjmOAkwBy7LWqmg7YdIpnfXLza+1unHrmjurkuM4BB/oRie5WniXMm/ALF1c1gAp7KjeQLdp73KVyLrvbzdEXsIPCOg/lVkLjHY4wBOMX5pPCB8hsNvLWLGH2rLM7aoYiRh9/Yoophi9NA1xDYUmCNwj7MYwcw11DXeNcDLcTedOlDsZ7mQ4+4adXX9raDcwsbS3k91nZwAfC8COndHITmKM+IqI2b1tGDL7Jnb/puFdN3Dd0DYEu4Z0DyPzy+IVPNEi9iiLyEiri4dDlPYqQxz+dZX0KSoHK+l9VlLxgJVPBLdXSsRIKeiy/+xnTyWjh6jcO4qP0Bjwp5z9RygM8f9C4XzWM928O3uoWRnZLnZatW4M67DYEg8iSMDBDELgCYFIDE1oDihFRCAuoEsiFJg2RMycSY4YipCRIwxZehER/OL2vAZIwFpmMHQHFzGOBEPEkhJHQEXIEhuQHGUgIQQSMMloJ0YtCxAPoMMixGGDhtJCQxsM5kEflFPECGJmLgkRDVBAUWhokXDDlkFk9g6LUhRgFJipwIvAiY4PYUaEmLEGIrwqGz2Au1J5NXjF4qiLatNOsEvXWX/bljvSWZlevdnBWsmm7e9BCJLRNue55DRJiS/muUxUDoXDhQkDhK5lbk6wXf+yLFrUhwB1Y8taViudNheqbWFWg/6Q1/Jctur2N5Bu+g1a1TZTz9UmzXWmZfFPiBGzhFkQ9Ynb0lKfuCMunJa0LOvs4q6BwEG3/1Z1CWRChClV7lyPmd74AmybVJooF9jKjXrjizgF6nowQN6qpkdsWZtD02FsOmfNmzLfDlWlLtq3smo3ta21gOpqs/XTYpkri6B1LFQt6VVS3l446Jhb68tdBT3sNpAs35Z5WSM4dlQIEOjaxLVWxuxskMJWBlsJ3PtCZ8NzElMrYdvEtVYKnOu21llKejMJ7tXoxpIFLD4OJRsZpgbaFLo97zutTq+2lhr53zfrBIKqmzZdkjzTkvPZThjNr1RdqNwFSwGe3JSbxkXvEIEv5ptGfZbt6rTI/q6WcOw+S8N8LSztRLc7zlSq1zDRjXfQSePWf8BW3WimlrXqLcxtceuAtU/xOHT3hu1Sv9Xl+qy4/gIxs7PV+ay3Z96kta5MaKIEqPhKbaMv040EIs/G88D4BqxIDakAkK0B8aMyOKkGwcO03NSyaFXjIblpV2Vti1o4r+ZIow8bXQA/QoSag5qrNVSzqLVxakN98NiprZONa1CZ/AH0sePRLZDw+N6YtdEt82olTS3doZLLO1VPcLLrfSyzXfTAOdZEoITKhUmllAswt1+4qWA5eyxHEWDd0aBbOKs+hex7t/BOsC+ALv50b07uPcEYa07rhP/c6I4rIRIdTt9B7M3PjxjxOXWIMR//eMTe/vyICZ9FDjFusXsGxNJyvZZFhgpbsn0u87tlWXjbYkFiczqRJCbkkKQGRwfSpu2fVzCLOJnUyUho2MJLnLZOxz3+cdp6D7h1pkmlhWrgCsimsW8NbZfj7M3fdJYpW+rMvu3ZEZZj1xLBrHMF6bLe1rfkEN8+HICNWpresJH0OyF4+EYPDMJtKFFfuEgifkQnJQyzcXUS+wRy/F5l8xjPQKjkJqrPCpMalU0m+8n0SqnK1DCfii+QUhrzpdI0i/51nOXx4Ex8JuLJZWFnftwxHgRbFPAx/D8Pzsnx4Aw5JApcRFOfsAnioUMaXIGjo0J3Sr4fdeYy1pR93zhm3afdD99j13H2+/Co7BdE1lGmSVzzdE8xn/dZP+Q/IIc9AOOQxHZhPDsExrOjgZH6jA0MTn4AjudwRnYwPHsoFLNvY2iO2wBR9jjSmL6gPguAAem4GAuX80zljqNRLuSPpwv1n8JNadyrp15XuU51ezDoHxzop3ugqwNAV0cCOhx6EvWvSUTcR9PmLyEseJbC43E+GAh+xxHKOSLbc8T7Qxjk/dEwCAO+CEgohiKvexuj3I9CTibj4JjYpxHhe355Js6u67K+P/G938P73SF4v/v/vvaNX95oyEYXFX25QqcF4o9IivfwSgfvuz14Lw/glctj4RXshyPOgCvsuD0SE9C7qptCiB8Fu58+5IXlAV5YHokXiB9OK29XoVAfs4kTuldM7vM4ikfjx8L0l84pyZ5TPh3CPJ+OhulDAi/zAeED1t2XSMAxkQgCtsM8gS+A9IPhEk8iotn4O2T7d5vuf268/i9QSwcI6DF16KkHAABpIgAAUEsBAhQAFAAIAAgAgWvMQtY3vbkZAAAAFwAAABYAAAAAAAAAAAAAAAAAAAAAAGdlb2dlYnJhX2phdmFzY3JpcHQuanNQSwECFAAUAAgACACBa8xC6DF16KkHAABpIgAADAAAAAAAAAAAAAAAAABdAAAAZ2VvZ2VicmEueG1sUEsFBgAAAAACAAIAfgAAAEAIAAAAAA==" /> <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.
Proof with finish.
intros.
assert_diffs.
Name I the midpoint of B and C.
Name J the midpoint of A and C.
elim (intersection_two_medians_exist A B C I J H H1 H4); intros G HG; spliter.
exists G; unfold unique; unfold is_gravity_center; repeat split...
exists I;  exists J; do 3 (split; finish).
intros G' HG'; spliter; decompose [ex and] H8;clear H8.
assert_all.
apply l6_21 with A x B x0...
intro;search_contradiction.
show_distinct' B x0...
Qed.
Ltac intersection_medians G A B C I J H1 H2 H3 :=
let T := fresh in assert(T:= intersection_two_medians_exist A B C I J H1 H2 H3);
ex_and T G.
Tactic Notation "Name" ident(G) "the" "intersection" "of" "the" "medians" "(" ident(A) ident(I) ")" "which" "is" "a" "median" "since" ident(H2) "and" "(" ident(B) ident(J) ")" "which" "is" "a" "median" "since" ident(H3) "of" "the" "non-flat" "triangle" ident(A) ident(B) ident(C) ident(H1) :=
intersection_medians G A B C I J H1 H2 H3.

Lemma three_medians_intersect:
forall A B C I J K,
~Col A B C ->
Midpoint I B C ->
Midpoint J A C ->
Midpoint K A B ->
exists G, Col G A I /\ Col G B J /\Col G C K.