Sehnenviereck SS 12: Unterschied zwischen den Versionen

Aktuelle Version vom 18. Juli 2012, 21:59 Uhr

Definitionen

Kreissehne

1. Es sei $\ k$ ein Kreis. Eine Sehne des Kreises ist jede Strecke, deren Anfangs- und Endpunkte Element des Kreises $\ k$ sind.

2. ..........

Durchmesser

1. Es sei $\ k$ ein Kreis mit dem Mittelpunkt $\ M$ . Ferner seien $\ A$ und $\ B$ zwei Punkte des Kreises $\ k$ . Ein Durchmesser ist die Strecke $\overline {AB}$ , für die gilt $\operatorname{Zw} \left( A, M, B\right)\wedge A,B\in \ k$ .

Radius

1. Es sei $\ k$ ein Kreis mit dem Mittelpunkt $\ M$ . Jede Strecke, die den Anfangspunkt in $\ M$ und den Endpunkt in einem beliebigen Punkt des Kreises $\ k$ hat, nennt man Radius.

Erarbeitung des Begriffs Sehnenviereck

Sehnenviereck

Ein Viereck, dessen Seiten Sehnen ein und desselben Kreises $k$ sind, heißt Sehnenviereck.

Sätze

Satzfindung

sehr speziell: Quadrate

Jedes Quadrat hat einen Umkreis und ist somit ein Sehnenviereck.

weniger speziell, aber immer noch ziemlich speziell: Rechtecke

Jedes Rechteck ist ein Sehnenviereck.

noch allgemeiner, aber immer noch ziemlich speziell: gleichschenklige Trapeze

Jedes gleichschenklige Trapez ist ein Sehnenviereck.

allgemeines Sehnenviereck

Ausgangslage: $\ \overline{ABCD}$ ist ein gleichschenkliges Trapez.

Arbeitsauftrag: Bewegen Sie den Punkt $\ C$ auf dem Kreis. Beobachten Sie, wie sich der rote und der blaue Winkel verändern. Was vermuten Sie bezüglich der Größe von $\ \gamma$ ? Was vermuten Sie hinsichtlich der Größen der gegenüberliegenden Winkel im Sehnenviereck?

Der Satz über die gegenüberliegenden Winkel im Sehnenviereck

Satz 1

In jedem Sehnenviereck sind die gegenüberliegenden Winkel supplementär.

--Oz44oz 20:32, 18. Jul. 2012 (CEST)

Satz 2 : Die Umkehrung vom Satz 1

Wenn in einem Viereck die gegenüberliegenden Winkel supplementär sind, dann ist das Viereck ein Sehnenviereck.

Kriterium

Ein Viereck ist .........

Beweise

wir wissen

--Oz44oz 22:55, 17. Jul. 2012 (CEST)

zu zeigen:

--Oz44oz 23:03, 17. Jul. 2012 (CEST)

Beweis vom Satz 1

Beweis 1	Beweis 2	Beweis 3

Beweisen Sie $\|\beta\|$ + $\|\delta\|$ = 180°	Beweisen Sie $\|\beta\|$ + $\|\delta\|$ = 180°	Beweisen Sie $\|\beta\|$ + $\|\delta\|$ = 180°

--Oz44oz 19:19, 16. Jul. 2012 (CEST)

Voraussetzung:

Behauptung:

Beweis 1:

Beweis vom Satz 2

Beweis 1	Beweis 2

Annahme: $D$ liegt ..	Annahme: $D$ liegt ..

--Oz44oz 19:15, 16. Jul. 2012 (CEST)

Voraussetzung:

Behauptung:

Annahme:

Beweis 1:

Funktionale Betrachtung

--Oz44oz 22:47, 16. Jul. 2012 (CEST)

--Oz44oz 22:45, 17. Jul. 2012 (CEST)

@@ Zeile 2: / Zeile 2: @@
-=== Sehne ===
+=== Kreissehne ===
+.   Es sei <math>\ k</math> ein Kreis. Eine Sehne des Kreises ist jede Strecke, deren Anfangs- und Endpunkte Element des Kreises <math>\ k</math> sind.
+.   ..........
+=== Durchmesser===
-=== Sehnenviereck ===
+.   Es sei <math>\ k</math> ein Kreis mit dem Mittelpunkt <math>\ M </math>. Ferner seien <math>\ A</math> und <math>\ B </math> zwei Punkte des Kreises <math>\ k</math>. Ein Durchmesser ist die Strecke <math>\overline {AB}</math>, für die gilt <math> \operatorname{Zw} \left( A, M, B\right)\wedge A,B\in \ k</math>.
+=== Radius ===
+.   Es sei <math>\ k</math> ein Kreis mit dem Mittelpunkt <math>\ M </math>. Jede Strecke, die den Anfangspunkt in <math>\ M </math> und den Endpunkt in einem beliebigen Punkt des Kreises <math>\ k</math> hat, nennt man Radius.
+=== Erarbeitung des Begriffs Sehnenviereck ===
+[[Datei:Sehnenvierecke.pdf]]
+=== Sehnenviereck ===
+Ein Viereck, dessen Seiten Sehnen ein und desselben Kreises <math>k</math> sind, heißt Sehnenviereck.
 == Sätze ==
@@ Zeile 17: / Zeile 31: @@
 ===Satzfindung===
-===Der Satz ===
+==== sehr speziell: Quadrate ====
+Jedes Quadrat hat einen Umkreis und ist somit ein Sehnenviereck.<br /><br />
+[[Bild:Quadrat_als_Sehnenviereck.png]]
+==== weniger speziell, aber immer noch ziemlich speziell: Rechtecke ====
+Jedes Rechteck ist ein Sehnenviereck.
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+==== noch allgemeiner, aber immer noch ziemlich speziell: gleichschenklige Trapeze ====
+Jedes gleichschenklige Trapez ist ein Sehnenviereck.
-====Satz 1 ====
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+==== allgemeines Sehnenviereck ====
+Ausgangslage: <math>\ \overline{ABCD}</math> ist ein gleichschenkliges Trapez.
+Arbeitsauftrag: Bewegen Sie den Punkt <math>\ C</math> auf dem Kreis. Beobachten Sie, wie sich der rote und der blaue Winkel verändern. Was vermuten Sie bezüglich der Größe von <math>\ \gamma</math>? Was vermuten Sie hinsichtlich der Größen der gegenüberliegenden Winkel im Sehnenviereck?
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+===Der Satz über die gegenüberliegenden Winkel im Sehnenviereck ===
+====Satz 1 ====
+'''In jedem Sehnenviereck sind die gegenüberliegenden Winkel supplementär.'''
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" showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />--[[Benutzer:Oz44oz|Oz44oz]] 20:32, 18. Jul. 2012 (CEST)
 ====Satz 2 : Die Umkehrung vom Satz 1====
+Wenn in einem Viereck die gegenüberliegenden Winkel supplementär sind, dann ist das Viereck ein Sehnenviereck.
 ===Kriterium ===
+Ein Viereck ist .........
 ==Beweise==
+===wir wissen===
+<ggb_applet width="1008" height="421"  version="4.0" 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showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
+--[[Benutzer:Oz44oz|Oz44oz]] 22:55, 17. Jul. 2012 (CEST)
+===zu zeigen:===
+<ggb_applet width="1008" height="421"  version="4.0" ggbBase64="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" showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
+--[[Benutzer:Oz44oz|Oz44oz]] 23:03, 17. Jul. 2012 (CEST)
 ===Beweis vom Satz 1===
 {| class="wikitable"
 !Beweis 1!!Beweis 2!!Beweis 3
@@ Zeile 42: / Zeile 92: @@
 | [[Datei:Sehnenviereck_Beweis_1.png| 300px]] || [[Datei:Sehnenviereck_Beweis_2.png| 300px]] || [[Datei:Sehnenviereck_Beweis_3.png| 300px]]
 |-
-| .. || .. || ..
+| Beweisen Sie <math>|\beta|</math> + <math>|\delta|</math> = 180°|| Beweisen Sie <math>|\beta|</math> + <math>|\delta|</math> = 180° || Beweisen Sie <math>|\beta|</math> + <math>|\delta|</math> = 180°
 |}
+--[[Benutzer:Oz44oz|Oz44oz]] 19:19, 16. Jul. 2012 (CEST)
+'''Voraussetzung:'''
+'''Behauptung:'''
+'''Beweis 1:'''
 ===Beweis vom Satz 2===
@@ Zeile 52: / Zeile 111: @@
 !Beweis 1!!Beweis 2
 |-
-| [[Datei:Sehnenviereck_Beweis_Umkehrung_1.png| 300px]] || [[Datei:Sehnenviereck_Beweis_Umkehrung_2.png| 300px]]]]
+| [[Datei:Sehnenviereck_Beweis_Umkehrung_1.png| 300px]] || [[Datei:Sehnenviereck_Beweis_Umkehrung_2.png| 300px]]
 |-
-| ..... || ........
+|Annahme: <math>D</math> liegt .. || Annahme: <math>D</math> liegt ..
 |}
 --[[Benutzer:Oz44oz|Oz44oz]] 19:15, 16. Jul. 2012 (CEST)
+'''Voraussetzung:'''
+'''Behauptung:'''
+'''Annahme:'''
+'''Beweis 1:'''
+===Funktionale Betrachtung===
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+--[[Benutzer:Oz44oz|Oz44oz]] 22:47, 16. Jul. 2012 (CEST)
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+--[[Benutzer:Oz44oz|Oz44oz]] 22:45, 17. Jul. 2012 (CEST)