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

In article <>, WM <> writes:
>On 5 Dez., 19:54, (Michael Stemper) wrote:
>> In article <>, WM <> writes:


>> >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


"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 says
that 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 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] <>
[2] <>
[3] <>

Michael F. Stemper
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