Degenerate Triangle

[Rade]

In another thread the topic was raised of the degenerate triangle. Below is a picture that was identified as being a degenerate triangle = three points connected by straight line segments.

 

o______o______o

 

Do all the mathematicians on the forum agree that this degenerate triangle is a triangle by definition ?

 

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Edit: Two other views of a degenerate triangle would be the case where two of the three points completely overlap at either end (note--not being able to show two points overlapping I place them next to each other as [oo]).

 

o___________[oo] or [oo]___________o

 

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Second Edit: My goal here is to better understand the important role of definition as relates to triangles. Clearly there are many ways to "define" a triangle. Some definitions lead to the possibility of the degenerate cases shown above, others exclude the degenerate case by definition. Which position is valid, or are they both valid ? Are there attributes present in what all would agree is a sound triangle not present in a degenerate triangle, something that would logically exclude the degenerate case from the set of all triangles ? Does the science of geometry recognize degenerate triangles as valid triangles, if so, how exactly are they used ?

 

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Third Edit: In my search I have run across this link: http://mathforum.org/library/drmath/view/64043.html--edit change to link

 

The author of this link, reaches the conclusion that the degenerate triangle cannot be seen to be within the set of the ordinary triangle by definition. The reason given is that the degenerate triangle does not lie in a unique plane, there are many planes for all the points. This plane criterion is outside the definition of a "triangle", thus, by definition, whatever the degenerate triangle is, it is not a triangle. It would appear that it requires its own concept label to identify the set of all such entities that do not lie in a unique plane, and have a mixture of triangle and non triangle characteristics.

 

Are there any comments on this interpretation of the degenerate triangle ?

Edited October 28, 2010 by Rade