Require Export GeoCoq.Highschool.triangles.
Require Export GeoCoq.Highschool.circumcenter.
Require Export GeoCoq.Highschool.gravityCenter.
(** In this file we give as example the proof of some exercises given in french high-school at 8th Grade (quatrième). The exercise is taken from http://mep-outils.sesamath.net/manuel_numerique/index.php?ouvrage=ms4_2011&page_gauche=169 *)
(** # <script type="text/javascript" src="https://www.geogebra.org/scripts/deployggb.js"></script> <script type="text/javascript"> var applet35 = new GGBApplet({material_id: "xXKV5j7J"}, true); var applet36 = new GGBApplet({material_id: "Pp4K9rkb"}, true); var applet37 = new GGBApplet({material_id: "pgjEKRMd"}, true); var applet38 = new GGBApplet({material_id: "rkDYJXyK"}, true); var applet39 = new GGBApplet({material_id: "wPE3EwBr"}, true); var applet40 = new GGBApplet({material_id: "FJ859DxG"}, true); var applet41 = new GGBApplet({material_id: "mPTQra6U"}, true); var applet42 = new GGBApplet({material_id: "d8PwNtGa"}, true); var applet44 = new GGBApplet({material_id: "N9SyfPGd"}, true); var applet45 = new GGBApplet({material_id: "dFfwvBut"}, true); var applet46 = new GGBApplet({material_id: "yV7nKC8E"}, true); var applet47 = new GGBApplet({material_id: "eKuUrjFE"}, true); var applet48 = new GGBApplet({material_id: "yV7nKC8E"}, true); var applet49 = new GGBApplet({material_id: "t2QJ3xbF"}, true); var applet50 = new GGBApplet({material_id: "WdJ6GvzE"}, true); var applet52 = new GGBApplet({material_id: "rAdHpavY"}, true);  window.onload = function() { applet35.inject('applet_container35', 'preferHTML5'); applet36.inject('applet_container36', 'preferHTML5'); applet37.inject('applet_container37', 'preferHTML5'); applet38.inject('applet_container38', 'preferHTML5'); applet39.inject('applet_container39', 'preferHTML5'); applet40.inject('applet_container40', 'preferHTML5'); applet41.inject('applet_container41', 'preferHTML5'); applet42.inject('applet_container42', 'preferHTML5'); applet44.inject('applet_container44', 'preferHTML5'); applet45.inject('applet_container45', 'preferHTML5'); applet46.inject('applet_container46', 'preferHTML5'); applet47.inject('applet_container47', 'preferHTML5'); applet48.inject('applet_container48', 'preferHTML5'); applet49.inject('applet_container49', 'preferHTML5'); applet50.inject('applet_container50', 'preferHTML5'); applet52.inject('applet_container52', 'preferHTML5');  } </script> # **)
Section Exercices.
Context `{TE:Tarski_euclidean}.
(** Exercice 35 : Dans un triangle rectangle a. GAZ est un triangle rectangle en A. Les points F, E et R sont les milieux respectifs de [AZ], [GZ] et [GA]. Fais une figure. b. Quelle est la nature du quadrilatère FERA ? Prouve-le. *)
(** # <div style="width:748px;height:397px;display:block" id="applet_container35"></div> # **)

Lemma sesamath_4ieme_G2_ex35 :
forall G A Z F E R,
~ Col G A Z ->
Per G A Z ->
Midpoint F A Z ->
Midpoint E G Z ->
Midpoint R G A ->
Rectangle F E R A.