Satz des Thales: Unterschied zwischen den Versionen

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(Geogebratest)
(Umkehrung Satz des Thales)
 
(28 dazwischenliegende Versionen von 9 Benutzern werden nicht angezeigt)
Zeile 1: Zeile 1:
=Test=
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=Ein wenig Didaktik=
 
Hier geben Ihnen die Didaktikspezialisten Tipps zum Satz des Thales
 
Hier geben Ihnen die Didaktikspezialisten Tipps zum Satz des Thales
  
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=Satzfindung=
  
==Funktionale Betrachtung==
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==Induktive Satzfindung==
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==Beweisführung==
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==Funktionale Betrachtung==
<ggb_applet width="1280" height="855"  version="3.2" ggbBase64="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" framePossible = "false" showResetIcon = "false" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" allowRescaling = "true" />
+
  
===Variante===
 
<ggb_applet width="884" height="620"  version="3.2" ggbBase64="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" framePossible = "false" showResetIcon = "true" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" allowRescaling = "true" />
 
--[[Benutzer:&quot;chris&quot;07|&quot;chris&quot;07]] 22:10, 14. Jul. 2010 (UTC)
 
  
=induktive Satzfindung=
+
===Variante 1===
<ggb_applet width="1280" height="636"  version="3.2" ggbBase64="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" framePossible = "false" showResetIcon = "true" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" allowRescaling = "true" />
+
<ggb_applet width="1005" height="544"  version="3.2" ggbBase64="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" framePossible = "false" showResetIcon = "true" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" allowRescaling = "true" />--[[Benutzer:&quot;chris&quot;07|&quot;chris&quot;07]] 21:47, 15. Jul. 2010 (UTC)
  
=Sehnen verschieben=
 
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--[[Benutzer:Mirasol|Mirasol]] 13:52, 9. Jul. 2010 (UTC)
 
Bewege den Punkt A, um die Sehne zu verschieben!
 
  
 
+
===Variante 2===
=Variante a=
+
  
 
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--[[Benutzer:&quot;chris&quot;07|&quot;chris&quot;07]] 21:12, 14. Jul. 2010 (UTC)
 
--[[Benutzer:&quot;chris&quot;07|&quot;chris&quot;07]] 21:12, 14. Jul. 2010 (UTC)
==Variante b==
+
 
 +
 
 +
===Variante 3===
 
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--[[Benutzer:&quot;chris&quot;07|&quot;chris&quot;07]] 21:12, 14. Jul. 2010 (UTC)
 
--[[Benutzer:&quot;chris&quot;07|&quot;chris&quot;07]] 21:12, 14. Jul. 2010 (UTC)
  
=induktive Satzfindung einer Umkehrung=
+
=Beweisfindung=
  
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(vgl. Idee von Frau "komplizierter Doppelname" in einer alten Klausur)<br />
+
==ikonisches/halbikonisches Beweisen==
Kann bitte jemand den Satz auf richtige Formulierung kontrollieren?! <br />
+
 
Welche speziellen Art der Einführung von Sätzen im induktiven Bereich lässt sich dieses Beispiel zuordnen?
+
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framePossible = "false" showResetIcon = "true" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" allowRescaling = "true" />--[[Benutzer:&quot;chris&quot;07|&quot;chris&quot;07]] 17:07, 15. Jul. 2010 (UTC)
=Funktional=
+
 
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+
==Beweisen am Beispiel==
 
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 +
 +
=induktive Satzfindung der allgemeinen Umkehrung=
 +
 +
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 +
 +
Nimmt man statt eines Rechtecks ein Parallelogramm, so lässt sich die Umkehrung des Periphriewinkelsatzes finden:<br />
 +
Die Scheitel kongruente Winkel, deren Schenkel die Eckpunkte einer Strecke AB enthalten, liegen auf einem Kreis, der AB als Sehne hat.--[[Benutzer:Tja???|Tja???]] 09:39, 23. Jul. 2010 (UTC)
 +
 +
=Beweisführung=
 +
 +
==Satz des Thales==
 +
===Satz des Thales===
 +
Es sei k ein Kreis mit einem Durchmesser <math> \overline {AB} </math>. Jeder Peripheriewinkel von k über <math> \overline {AB} </math> ist ein rechter Winkel.--[[Benutzer:Löwenzahn|Löwenzahn]] 15:07, 23. Jul. 2010 (UTC)
 +
 +
Ein Versuch den Satz des Thales  mit dem EP zu beweisen:
 +
 +
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 +
 +
Vor: Dreieck ABC, A,B und C element von k, Kreis K<br />
 +
Beh: y= 90<br />
 +
 +
1) Konstruieren eine Parallele zu AB durch C  (Nach dem EP)<br />
 +
2)Der Winkel < ACE ist Kongruent zu alpha  (Wechselwinkelsatz)<br />
 +
3)Delta ist kongruent zu Betha  (Wechselwinkelsatz)<br />
 +
4)Winkel < ACM ist Kongruent zu alpha  (Basiswinkelsatz)<br />
 +
5) Winkel < MCB ist kongruent zu Betha  (Basiswinkelsatz)<br />
 +
6) <ACE+<ACM+<MCB+<BCD= 180<br />
 +
7)alpha+alpha+betha+Betha= 180  (einsetzten der Kongruenzen)<br />
 +
8) 2*(alpha+betha)= 180    (rechenen in R)<br />
 +
9) alpha+betha=90            (rechenen in R)<br />
 +
10) Y=90<br />
 +
q.e.d<br />
 +
 +
==Umkehrung 1: Satz des Thales==
 +
===Umkehrung Satz des Thales===
 +
Ist <math> \overline {ABC} </math> ein Dreieck mit einem rechten WInkel bei <math> C </math>, so liegt der Punkt <math> C </math> auf dem Thaleskreis, wobei <math> \overline {AB} </math> einen Durchmesser des Kreises <math> k </math>bildet.--[[Benutzer:Löwenzahn|Löwenzahn]] 15:07, 23. Jul. 2010 (UTC)<br />
 +
Anmerkung--[[Benutzer:TimoRR|TimoRR]] 16:13, 26. Jul. 2010 (UTC): Du meinst <math>\overline{AB}</math> ist Durchmesser des Kreises <math>k</math>. !?? Jupp, Danke!--[[Benutzer:Löwenzahn|Löwenzahn]] 16:25, 26. Jul. 2010 (UTC)
 +
 +
==Umkehrung 2: Satz des Thales==
 +
===Umkehrung Satz des Thales===
 +
 +
<br />Ist ein Peripheriewinkel <math>\gamma </math> über einer Sehne <math> s </math> eines Kreises <math> k </math> ein rechter Winkel, so ist die Sehne s ein Durchmesser des Kreises <math> k </math>.--[[Benutzer:Löwenzahn|Löwenzahn]] 15:07, 23. Jul. 2010 (UTC)
 +
 +
<br />Ein [[Beweise_von_Studenten#Umkehrung_des_Satz_des_Thales | Versuch eines Beweises]] besser: zwei Beweis-Ideen, eine über Winkelkonstruktion, die andere via Zentri-Peripheriewinkelsatz...
 +
<br />--[[Benutzer:Heinzvaneugen|Heinzvaneugen]] 10:29, 26. Jul. 2010 (UTC)
 +
 +
===Kommentar zu den Umkehrungen des Thalesstzes--[[Benutzer:*m.g.*|*m.g.*]] 20:43, 23. Jul. 2010 (UTC)===
 +
Es sei <math>\ \alpha</math> ein Winkel und <math>\ k</math> ein Kreis.
 +
Der Satz des Thales hat zwei Voraussetzungen:
 +
 +
# <math>\ \alpha</math> ist Peripheriewinkel von <math>\ k</math>
 +
# über einem Durchmesser von <math> \ k</math>.
 +
 +
Die Behauptung des Thalessatzes: <math>\ \alpha</math> ist ein rechter Winkel.
 +
 +
Aus Gründen der Übersicht benenne ich die Voraussetzungen V1 und V2. Für die Behauptung schreibe ich B.
 +
 +
Satz des Thales:
 +
 +
Aus V1 und V2 folgt B.
 +
 +
Die eigentliche Umkehrung:
 +
 +
Aus B folgt V1 und V2.
 +
 +
Gemischte Umkehrung 1:
 +
 +
Aus B und V1 folgt V2.
 +
 +
Gemischte Umkehrung 2:
 +
 +
Aus B und V2 folgt V1.
 +
 +
<br /><br />Also fehlt uns noch die
 +
 +
==== Eigentliche Umkehrung des Satz von Thales ====
 +
Mein Vorschlag: <br />Es sei ein Dreieck <math> \overline {ABC} </math> mit den schulüblichen Bezeichnungen. Ist <math>\ \gamma</math> ein rechter Winkel, so ist <math> \ c</math>  identisch mit einem Durchmesser des Umkreises des Dreiecks.
 +
<br />--[[Benutzer:Barbarossa|Barbarossa]] 08:38, 25. Jul. 2010 (UTC)

Aktuelle Version vom 26. Juli 2010, 17:25 Uhr

Inhaltsverzeichnis

Ein wenig Didaktik

Hier geben Ihnen die Didaktikspezialisten Tipps zum Satz des Thales

Satzfindung

Induktive Satzfindung

--Gubbel 12:10, 21. Jul. 2010 (UTC)

Funktionale Betrachtung

Variante 1

--"chris"07 21:47, 15. Jul. 2010 (UTC)


Variante 2

--"chris"07 21:12, 14. Jul. 2010 (UTC)


Variante 3

--"chris"07 21:12, 14. Jul. 2010 (UTC)

Beweisfindung

ikonisches/halbikonisches Beweisen

--"chris"07 17:07, 15. Jul. 2010 (UTC)

Beweisen am Beispiel

induktive Satzfindung der allgemeinen Umkehrung

Nimmt man statt eines Rechtecks ein Parallelogramm, so lässt sich die Umkehrung des Periphriewinkelsatzes finden:
Die Scheitel kongruente Winkel, deren Schenkel die Eckpunkte einer Strecke AB enthalten, liegen auf einem Kreis, der AB als Sehne hat.--Tja??? 09:39, 23. Jul. 2010 (UTC)

Beweisführung

Satz des Thales

Satz des Thales

Es sei k ein Kreis mit einem Durchmesser  \overline {AB} . Jeder Peripheriewinkel von k über  \overline {AB} ist ein rechter Winkel.--Löwenzahn 15:07, 23. Jul. 2010 (UTC)

Ein Versuch den Satz des Thales mit dem EP zu beweisen:

Vor: Dreieck ABC, A,B und C element von k, Kreis K
Beh: y= 90

1) Konstruieren eine Parallele zu AB durch C (Nach dem EP)
2)Der Winkel < ACE ist Kongruent zu alpha (Wechselwinkelsatz)
3)Delta ist kongruent zu Betha (Wechselwinkelsatz)
4)Winkel < ACM ist Kongruent zu alpha (Basiswinkelsatz)
5) Winkel < MCB ist kongruent zu Betha (Basiswinkelsatz)
6) <ACE+<ACM+<MCB+<BCD= 180
7)alpha+alpha+betha+Betha= 180 (einsetzten der Kongruenzen)
8) 2*(alpha+betha)= 180 (rechenen in R)
9) alpha+betha=90 (rechenen in R)
10) Y=90
q.e.d

Umkehrung 1: Satz des Thales

Umkehrung Satz des Thales

Ist  \overline {ABC} ein Dreieck mit einem rechten WInkel bei  C , so liegt der Punkt  C auf dem Thaleskreis, wobei  \overline {AB} einen Durchmesser des Kreises  k bildet.--Löwenzahn 15:07, 23. Jul. 2010 (UTC)
Anmerkung--TimoRR 16:13, 26. Jul. 2010 (UTC): Du meinst \overline{AB} ist Durchmesser des Kreises k. !?? Jupp, Danke!--Löwenzahn 16:25, 26. Jul. 2010 (UTC)

Umkehrung 2: Satz des Thales

Umkehrung Satz des Thales


Ist ein Peripheriewinkel \gamma über einer Sehne  s eines Kreises  k ein rechter Winkel, so ist die Sehne s ein Durchmesser des Kreises  k .--Löwenzahn 15:07, 23. Jul. 2010 (UTC)


Ein Versuch eines Beweises besser: zwei Beweis-Ideen, eine über Winkelkonstruktion, die andere via Zentri-Peripheriewinkelsatz...
--Heinzvaneugen 10:29, 26. Jul. 2010 (UTC)

Kommentar zu den Umkehrungen des Thalesstzes--*m.g.* 20:43, 23. Jul. 2010 (UTC)

Es sei \ \alpha ein Winkel und \ k ein Kreis. Der Satz des Thales hat zwei Voraussetzungen:

  1. \ \alpha ist Peripheriewinkel von \ k
  2. über einem Durchmesser von  \ k.

Die Behauptung des Thalessatzes: \ \alpha ist ein rechter Winkel.

Aus Gründen der Übersicht benenne ich die Voraussetzungen V1 und V2. Für die Behauptung schreibe ich B.

Satz des Thales:

Aus V1 und V2 folgt B.

Die eigentliche Umkehrung:

Aus B folgt V1 und V2.

Gemischte Umkehrung 1:

Aus B und V1 folgt V2.

Gemischte Umkehrung 2:

Aus B und V2 folgt V1.



Also fehlt uns noch die

Eigentliche Umkehrung des Satz von Thales

Mein Vorschlag:
Es sei ein Dreieck  \overline {ABC} mit den schulüblichen Bezeichnungen. Ist \ \gamma ein rechter Winkel, so ist  \ c identisch mit einem Durchmesser des Umkreises des Dreiecks.
--Barbarossa 08:38, 25. Jul. 2010 (UTC)