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If a ray r emanating from an exterior point of triangle ABC intersects side AB at any point between A and B, then r also intersects side AC or side BC [duplicate]

Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of themGiven triangle ABC, if D is an interior pt and E is an exterior pt, segment DE intersects triangle ABC?Finding the Perimiter of a right Triangle given an interior angle bisector and exterior angle bisectorQuestion about proof regarding a triangle and a line.Given a point origin, find ray that intersects two linesProve In Equilateral Triangle $ABC$ $AM$ and $AN$ are equalProve that ray CE intersects $triangle ABC$ at a point $D$ on $AB$ and that $D$ must lie strictly between $A$ and $B$.Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of themGeometry: Construct a rectangle with area equal to a given triangle and with one side equal to a given segment.Given a triangle $ABC$, make it a point $D$ on the side $AB$.Prove that if a ray has its endpoint on a line, where the ray is not on the line, then the points of the ray are on the same side of the line.

1

$begingroup$

This question is an exact duplicate of:

• 
  Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of them
  
  1 answer
  
  

Prove Proposition 3.9:
If a ray r emanating from an exterior point of triangle $ABC$ intersects side $AB$ at any point between $A$ and $B$, then $r$ also intersects side $AC$ or side $BC$.

Can someone help me finish this proof I started

By the hypothesis, let $DE$ be a ray emanating from an exterior point $D$ of triangle $ABC$ where the ray $DE$ intersects $AB$ at point $E$ such that $A*E*B$.

By pachs theorem, the line DE either intersects $AC$ or $BC$.
3.Suppose $DE$ intersects $AC$ at point $M$ where $M$ is not equal to $A$
Then by the definition of line segment, either $M=C$, or $D*M*E$

geometry euclidean-geometry education math-history

share|cite|improve this question

edited Mar 15 at 17:45

Math1234

asked Mar 15 at 17:05

Math1234Math1234

63

$endgroup$

marked as duplicate by Lee David Chung Lin, Leucippus, Lord Shark the Unknown, Vinyl_cape_jawa, uniquesolution Mar 16 at 9:57

This question was marked as an exact duplicate of an existing question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Please change the title of your question - proposition 3.9a is not something people will know.
  $endgroup$
  – Piotr Benedysiuk
  Mar 15 at 17:18
  

  
  
  
  

add a comment | 

1

$begingroup$

This question is an exact duplicate of:

• 
  Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of them
  
  1 answer
  
  

Prove Proposition 3.9:
If a ray r emanating from an exterior point of triangle $ABC$ intersects side $AB$ at any point between $A$ and $B$, then $r$ also intersects side $AC$ or side $BC$.

Can someone help me finish this proof I started

By the hypothesis, let $DE$ be a ray emanating from an exterior point $D$ of triangle $ABC$ where the ray $DE$ intersects $AB$ at point $E$ such that $A*E*B$.

By pachs theorem, the line DE either intersects $AC$ or $BC$.
3.Suppose $DE$ intersects $AC$ at point $M$ where $M$ is not equal to $A$
Then by the definition of line segment, either $M=C$, or $D*M*E$

geometry euclidean-geometry education math-history

share|cite|improve this question

edited Mar 15 at 17:45

Math1234

asked Mar 15 at 17:05

Math1234Math1234

63

$endgroup$

marked as duplicate by Lee David Chung Lin, Leucippus, Lord Shark the Unknown, Vinyl_cape_jawa, uniquesolution Mar 16 at 9:57

This question was marked as an exact duplicate of an existing question.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Please change the title of your question - proposition 3.9a is not something people will know.
  $endgroup$
  – Piotr Benedysiuk
  Mar 15 at 17:18
  

  
  
  
  

add a comment | 

$begingroup$

This question is an exact duplicate of:

• 
  Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of them
  
  1 answer
  
  

Prove Proposition 3.9:
If a ray r emanating from an exterior point of triangle $ABC$ intersects side $AB$ at any point between $A$ and $B$, then $r$ also intersects side $AC$ or side $BC$.

Can someone help me finish this proof I started

By the hypothesis, let $DE$ be a ray emanating from an exterior point $D$ of triangle $ABC$ where the ray $DE$ intersects $AB$ at point $E$ such that $A*E*B$.

By pachs theorem, the line DE either intersects $AC$ or $BC$.
3.Suppose $DE$ intersects $AC$ at point $M$ where $M$ is not equal to $A$
Then by the definition of line segment, either $M=C$, or $D*M*E$

geometry euclidean-geometry education math-history

share|cite|improve this question

edited Mar 15 at 17:45

Math1234

asked Mar 15 at 17:05

Math1234Math1234

63

$endgroup$

share|cite|improve this question

edited Mar 15 at 17:45

Math1234

asked Mar 15 at 17:05

Math1234Math1234

63

share|cite|improve this question

edited Mar 15 at 17:45

Math1234

asked Mar 15 at 17:05

Math1234Math1234

63

asked Mar 15 at 17:05

Math1234Math1234

63

asked Mar 15 at 17:05

Math1234Math1234

63

1 Answer
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Then by the definition of line segment, either $M=C$, or $D*M*E$

This step is not correct/justified.

Up to that point your proof is correct. In fact, the only thing you need to conclude the proof is that $D*E*M$ or $D*M*E$. Since these points are distinct, you may assume the contrary i.e. $E*D*M$, and show that $D$ is an interior point of a triangle which contradicts the assumption. To do this, prove that $D$ belongs to all three halfplanes which determine the interior of a triangle.

share|cite|improve this answer

answered Mar 15 at 19:13

KulistyKulisty

419216

$endgroup$

add a comment | 

0

$begingroup$

Then by the definition of line segment, either $M=C$, or $D*M*E$

This step is not correct/justified.

Up to that point your proof is correct. In fact, the only thing you need to conclude the proof is that $D*E*M$ or $D*M*E$. Since these points are distinct, you may assume the contrary i.e. $E*D*M$, and show that $D$ is an interior point of a triangle which contradicts the assumption. To do this, prove that $D$ belongs to all three halfplanes which determine the interior of a triangle.

share|cite|improve this answer

answered Mar 15 at 19:13

KulistyKulisty

419216

$endgroup$

add a comment | 

0

$begingroup$

Then by the definition of line segment, either $M=C$, or $D*M*E$

This step is not correct/justified.

Up to that point your proof is correct. In fact, the only thing you need to conclude the proof is that $D*E*M$ or $D*M*E$. Since these points are distinct, you may assume the contrary i.e. $E*D*M$, and show that $D$ is an interior point of a triangle which contradicts the assumption. To do this, prove that $D$ belongs to all three halfplanes which determine the interior of a triangle.

share|cite|improve this answer

answered Mar 15 at 19:13

KulistyKulisty

419216

$endgroup$

add a comment | 

$begingroup$

Then by the definition of line segment, either $M=C$, or $D*M*E$

This step is not correct/justified.

Up to that point your proof is correct. In fact, the only thing you need to conclude the proof is that $D*E*M$ or $D*M*E$. Since these points are distinct, you may assume the contrary i.e. $E*D*M$, and show that $D$ is an interior point of a triangle which contradicts the assumption. To do this, prove that $D$ belongs to all three halfplanes which determine the interior of a triangle.

share|cite|improve this answer

answered Mar 15 at 19:13

KulistyKulisty

419216

$endgroup$

share|cite|improve this answer

answered Mar 15 at 19:13

KulistyKulisty

419216

answered Mar 15 at 19:13

KulistyKulisty

419216

answered Mar 15 at 19:13

KulistyKulisty

419216