
Re: Re: Matheology § 170
Posted:
Dec 6, 2012 12:56 PM


In article <8fdbfffe9ad44409a70af26497c59753@o6g2000yql.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> writes: >On 5 Dez., 19:54, mstem...@walkabout.empros.com (Michael Stemper) wrote: >> In article <0e30135801064609b62814da5781d...@4g2000yql.googlegroups.com>, WM <mueck...@rz.fhaugsburg.de> writes:
>1 >11 >111 >> >> >In mathematics a triangle is defined by one angle and its two sides. >> >> No, in mathematics a triangle is defined by either its three vertices or >> its three sides. Two rays with a common endpoint define an angle, but not >> a triangle. > >Two *sides* with an angle defined by these sides define a triangle.
Repeating a lie doesn't make it true.
From Wikipedia[1]:
"A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments."
and:
"In Euclidean geometry any three points, when noncollinear, determine a unique triangle [...]"
Both of these refer to the need for three vertices; neither of them says that one is enough.
From MathWorld[2]:
"A triangle is a 3sided polygon [...]"
What do they say a polygon is?[3]
"A polygon can be defined (as illustrated above) as a geometric object 'consisting of a number of points (called vertices) and an equal number of line segments (called sides), [...]'"
So, they require that a triangle have three vertices, and also point out that its sides must be line *segments*.
They also provide these alternative definitions:
"There is unfortunately substantial disagreement over the definition of a polygon. Other sources commonly define a polygon (in the sense illustrated above) as a 'closed plane figure with straight edges' (Gellert et al. 1989, p. 162), 'a closed plane figure bounded by straight line segments as its sides' (Bronshtein et al. 2003, p. 137), or 'a closed plane figure bounded by three or more line segments that terminate in pairs at the same number of vertices, and do not intersect other than at their vertices' (Borowski and Borwein 2005, p. 573)."
Something with rays, as WM proposes, would not be a "closed plane figure", which all of these competing definitions require.
So, two rays and the angle between them do *not* define a triangle.
[1] <http://en.wikipedia.org/wiki/Triangle> [2] <http://mathworld.wolfram.com/Triangle.html> [3] <http://mathworld.wolfram.com/Polygon.html>
 Michael F. Stemper #include <Standard_Disclaimer> 91.2% of all statistics are made up by the person quoting them.

