```Date: Dec 6, 2012 12:56 PM
Author: Michael Stemper
Subject: Re: Re: Matheology § 170

In article <8fdbfffe-9ad4-4409-a70a-f26497c59753@o6g2000yql.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> writes:>On 5 Dez., 19:54, mstem...@walkabout.empros.com (Michael Stemper) wrote:>> In article <0e301358-0106-4609-b628-14da5781d...@4g2000yql.googlegroups.com>, WM <mueck...@rz.fh-augsburg.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 non-collinear, determine  a unique triangle [...]"Both of these refer to the need for three vertices; neither of them saysthat one is enough.From MathWorld[2]:  "A triangle is a 3-sided 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 outthat 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.
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