It is known that NOT all triangles on the hyperbolic plane have a circle that contains the triangle and passes thru all its 3 vertices. IOW, the circumcircle is not a universal property of triangles.
What about the incircle? If every triangle does have a unique largest circle contained within it, how do we characterize this circle - for example, will the center of the incircle always be the intersection of angle bisectors?
Yes, in hyperbolic geometry all triangles have an incircle, whose center lies on all three angle bisectors of the triangle. The radius of this incircle is always at most $\tanh^{-1} (1/2) \approx 0.5493$.