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Topic: What is a triangle?

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Subject:   RE: What is a triangle?
Author: Mathman
Date: Jan 2 2005
On Jan  2 2005, rabeldin wrote:
> The name "triangle" suggests a figure with three angles. This
> definition excludes U shaped figures with two parallel sides since
> they have but two angles. However, if you ask most people for the
> definition, they will respond in terms like "a three sided polygon"
> or "a closed figure with three sides". We seem to prefer a
> definition in terms of sides as easier than one in terms of angles.
> What we call triangles could legitimately be called "trigons",
> polygons with three sides. Are triangles and trigons completely
> equivalent? If we generalize to a non-plane space, they may be
> different. At least the question bears investigation.

I suspect
> that there are subtle errors in school mathematics where we mentally
> substitute an informal definition for the formal one and generalize
> improperly. Has anyone spent any time on such issues?

Sorry, but, unless I'm reading you wrongly, there is no issue.  The triangle is
a "closed" polygon with three sides, or accordingly three angles, being the same
in number.  It is the closed polygon with the least number of sides, sharing
properties of all closed polygons.  Definition requires that the information be
"necessary and sufficient."  In fact, three distinct, non-collinear points
also determine a triangle in 2-space or higher.  Your U-shape is determined
by [and requires] four points, for example.

A "closed figure with three sides" is necessary and sufficient, "sides" implying
straight lines.  A "polygon with three sides" is necessary, but not sufficient,
since it must be also "closed".  

Then we have non-Euclidean definitions, but I'm sure you mean Euclidean
geometry here.


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