Sehnenviereck SS 12: Unterschied zwischen den Versionen

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(Satzfindung)
(Funktionale Betrachtung)
 
(34 dazwischenliegende Versionen von 2 Benutzern werden nicht angezeigt)
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=== Sehne ===
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=== Kreissehne ===
  
  
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1.  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.
  
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2.  ..........
  
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=== Durchmesser===
  
=== Sehnenviereck ===
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1.  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>.
  
  
  
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=== Radius ===
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1.  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.
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=== Erarbeitung des Begriffs Sehnenviereck ===
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[[Datei:Sehnenvierecke.pdf]]
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=== Sehnenviereck ===
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Ein Viereck, dessen Seiten Sehnen ein und desselben Kreises <math>k</math> sind, heißt Sehnenviereck.
  
 
== Sätze ==
 
== Sätze ==
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===Satzfindung===
 
===Satzfindung===
  
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==== sehr speziell: Quadrate ====
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Jedes Quadrat hat einen Umkreis und ist somit ein Sehnenviereck.<br /><br />
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[[Bild:Quadrat_als_Sehnenviereck.png]]
  
  
  
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==== weniger speziell, aber immer noch ziemlich speziell: Rechtecke ====
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Jedes Rechteck ist ein Sehnenviereck.
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showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
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==== noch allgemeiner, aber immer noch ziemlich speziell: gleichschenklige Trapeze ====
--[[Benutzer:Oz44oz|Oz44oz]] 22:45, 17. Jul. 2012 (CEST)
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Jedes gleichschenklige Trapez ist ein Sehnenviereck.
  
===Der Satz ===
+
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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 ====
 
  
  
  
  
 +
====Satz 1 ====
  
 +
'''In jedem Sehnenviereck sind die gegenüberliegenden Winkel supplementär.'''
 +
<ggb_applet width="1008" height="411"  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]] 20:32, 18. Jul. 2012 (CEST)
  
 
====Satz 2 : Die Umkehrung vom Satz 1====
 
====Satz 2 : Die Umkehrung vom Satz 1====
 +
 +
Wenn in einem Viereck die gegenüberliegenden Winkel supplementär sind, dann ist das Viereck ein Sehnenviereck.
  
 
===Kriterium ===
 
===Kriterium ===
 +
 +
Ein Viereck ist .........
  
 
==Beweise==
 
==Beweise==
===1===
+
===wir wissen===
 +
 
  
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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)
 
--[[Benutzer:Oz44oz|Oz44oz]] 22:55, 17. Jul. 2012 (CEST)
  
===2===
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===zu zeigen:===
  
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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)
 
--[[Benutzer:Oz44oz|Oz44oz]] 23:03, 17. Jul. 2012 (CEST)
  
Zeile 60: Zeile 92:
 
| [[Datei:Sehnenviereck_Beweis_1.png| 300px]] || [[Datei:Sehnenviereck_Beweis_2.png| 300px]] || [[Datei:Sehnenviereck_Beweis_3.png| 300px]]
 
| [[Datei:Sehnenviereck_Beweis_1.png| 300px]] || [[Datei:Sehnenviereck_Beweis_2.png| 300px]] || [[Datei:Sehnenviereck_Beweis_3.png| 300px]]
 
|-  
 
|-  
| Beweisen Sie.. || Beweisen Sie.. || Beweisen Sie..
+
| 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)
 
--[[Benutzer:Oz44oz|Oz44oz]] 19:19, 16. Jul. 2012 (CEST)
Zeile 81: Zeile 113:
 
| [[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]]
 
|-  
 
|-  
| Beweisen Sie..... || Beweisen Sie........
+
|Annahme: <math>D</math> liegt .. || Annahme: <math>D</math> liegt ..
 
|}
 
|}
 
--[[Benutzer:Oz44oz|Oz44oz]] 19:15, 16. Jul. 2012 (CEST)
 
--[[Benutzer:Oz44oz|Oz44oz]] 19:15, 16. Jul. 2012 (CEST)
Zeile 94: Zeile 126:
  
 
'''Beweis 1:'''
 
'''Beweis 1:'''
 +
 +
===Funktionale Betrachtung===
 +
 +
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showResetIcon = "false" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
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--[[Benutzer:Oz44oz|Oz44oz]] 22:47, 16. Jul. 2012 (CEST)
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 +
 +
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 +
--[[Benutzer:Oz44oz|Oz44oz]] 22:45, 17. Jul. 2012 (CEST)

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

Inhaltsverzeichnis

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

Sehnenvierecke.pdf

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.

Quadrat als Sehnenviereck.png


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
Sehnenviereck Beweis 1.png Sehnenviereck Beweis 2.png Sehnenviereck Beweis 3.png
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
Sehnenviereck Beweis Umkehrung 1.png Sehnenviereck Beweis Umkehrung 2.png
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)