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

$\gamma_{1}(0)=A$;

$\gamma_{1}(x_{1})=B$;

$\gamma_{2}(0)=A$;

$\gamma_{2}(x_{2})=C$;

$\gamma_{3}(0)=B$;

$\gamma_{3}(x_{3})=C$.
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.
This entry is not yet complete, as a geodesic metric space has not yet been defined on PlanetMath. If the words “geodesic metric space” are clickable in the previous sentence, please let me know right away. Thanks.
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