How to define a triangle

I was asked is there a standard definition of triangle in geometry. The only I can think is that three points which are not on the same line joined by line segments but are there another as not all geometries are for example Euclidean? And on the other hand, for every positive integer $n$ there are different spaces $\mathbb{R}^n$ so we have trivially different definitions in every such a space if we count "Pick $x,y,z\in\mathbb{R}^2$ ..." as a different from the definition "Pick $x,y,z\in\mathbb{R}^3$ ...". But is that the only difference in definitions?

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It seems to me that you are suggesting that what you may be searching for is whether the conditions for a triangle involve three distinct points and a metric, or simply three distinct points and edges between them without a notion of distance (metric). Projective geometries allow for triangles without metrics. –  Mark Bennet Sep 19 '12 at 21:38
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2 Answers

On a Riemannian manifold a natural generalization is to geodesic triangles, that is, a triple of points connected by three geodesics. On a sphere we get spherical triangles, on the hyperbolic plane we get hyperbolic triangles, etc.

There is no particular reason to exclude degenerate triangles.

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The only I can think is that three points which are not on the same line joined by line segments but are there another as not all geometries are for example Euclidean?

Here is the difference between Euclidean Geometry (your definition) and Projective Geometry's definition:

"A triangle is a set of three non collinear points and the three lines incident with each pair of these points. The points are called vertices, and the lines are called sides. A triangle in a projective geometry is different from a triangle in Euclidean geometry. Each side of a triangle in projective geometry is a line, whereas each side of a triangle in Euclidean geometry is a segment. Betweenness of points in projective geometry is not defined; therefore, projective geometry does not have segments defined.".

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