On a 2-dimensional sphere of radius $\frac{1}{\sqrt{k}}$, call it $S_k$, where $k > 0$, we have the metric $d$ that is the great circle distance between any two points. How do I prove the following?
If $a + b + c < \frac{2 \pi}{\sqrt{k}}$, then there exists a triangle on $S_k$ having sides that are arcs of great circles with lengths $a$, $b$ and $c$. Also it is unique upto rigid motions.
A rigid motion is an isometry from $S_k$ onto itself.
Thanks.