Below we find the another triangle ABC, this time within the contraints of hyperbolic geometry. Whew, there is a lot going on down there. The sides of the triangle are the lines in bold and the red lines show the perpendicular bisectors of each side. There is a lot to say about this particular construction. To start, notice that an intersection exists between the three perpendicular bisectors.
Now, we cannot say that IA = IB = IC as we did in the last one (in this case, 'I' now denotes our intersection). As we move our vertices around, we see these values change constantly, and even as they are initially set up, they are still not equal. It is also possible to construct a situation in which no intersection exists, as the lines in hyperbolic geometry bend away from each other (Move the points and see for yourself!).