geodesic triangle
Let $M$ be a differentiable manifold (at least two times differentiable^{}) and $A,B,C\in M$ (not necessarily distinct). Let ${x}_{1},{x}_{2},{x}_{3}\in [0,\mathrm{\infty})$. Let ${\gamma}_{1}:[0,{x}_{1}]\to M$, ${\gamma}_{2}:[0,{x}_{2}]\to M$, and ${\gamma}_{3}:[0,{x}_{3}]\to M$ be geodesics such that all of the following hold:

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${\gamma}_{1}(0)=A$;

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${\gamma}_{1}({x}_{1})=B$;

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${\gamma}_{2}(0)=A$;

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${\gamma}_{2}({x}_{2})=C$;

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${\gamma}_{3}(0)=B$;

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${\gamma}_{3}({x}_{3})=C$.
Then the figure determined by ${\gamma}_{1}$, ${\gamma}_{2}$, and ${\gamma}_{3}$ is a geodesic triangle.
Note that a geodesic triangle need not be a triangle. For example, in ${\mathbb{R}}^{2}$, if $A=(0,0)$, $B=(1,2)$, and $C=(3,6)$, then the geodesic triangle determined by $A$, $B$, and $C$ is $\{(x,2x):x\in [0,3]\}$, which is not a triangle.
geodesic metric space (http://planetmath.org/GeodesicMetricSpace)
Title  geodesic triangle 

Canonical name  GeodesicTriangle 
Date of creation  20130322 17:11:31 
Last modified on  20130322 17:11:31 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  7 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 53C22 