The incenter is defined a point inside a triangle that is equal distance from each side. It can be constructed by constructing the angle bisectors of the three angles and locating the point of intersection.

A synopsis of the proof has you construct the incenter from the intersection of two of the angles and then show that the segment from the incenter to the point without an angle bisector is indeed an angle bisector. This proof uses the AAS congruency criterion for triangles to that there are three pairs of congruent triangles formed from the incenter and this shows that the segment from the incenter to the remaining point is an angle bisector. The AAS criteria is dependent on the Alternative Opposite Angle Theorem which is equivalent to the fifth postulate. Thus the proof of the incenter uses the fifth postulate.

The hyperbolic incenter exists for all triangles. The angle bisectors of two sides of a triangle is the points that are equal distance from both sides. The intersection of three angle bisectors is the point that is equal distance from all three sides which is one of Euclid's common notions.

You can not inscribe a taxicab circle in this triangle since only two points are touching. It is possible to find triangles that you can inscribe taxicab circles in. The construction in this applet is robust so moving around the points A, B and C you can find the triangles that it is possible to inscribe a taxicab circle with the same center as the Euclidean incenter.