Consider the following figure of a triangle within a triangle in the hyperbolic plane (I have not tried to make the geodesics look like semi-circular arcs).
Here $PY', PX', X'Y'$, and $XY$ are geodesic segments. The lengths of $PX, PY, PX', PY', XY$, and $X'Y'$ are $a, a, 2a, 2a, b$, and $b'$ respectively, as shown in the figure.
I want to show that $b'>2b$, using Jacobi fields.
The reason I want to use Jacobi fields since "Jacobi fields measure the rate of spreading of geodesics emanating from a point". But I fail to see how to really put this into practice.
Another reason I think Jacobi fields can be used was a remark I read somewhere that it can be proved using Jacobi fields that the hyperbolic plane is a $CAT(0)$ space.