Hyperbolic circumference: center and point

To make the construction we will suppose that the point does not belong to the perpendicular line to the boundary line. We also suppose that the first given point is the center and the second the point from which we want  the circumference to pass. The steps to be done are the following ones:

1. Plot the perpendicular line to the boundary line that goes through the center, C, (the first point).
2. Plot the circumference with center in the boundary line and that goes through C and D. We will always be able to plot this circumference since we suppose that C and D do not belong in the same perpendicular line to the boundary line, where it is the center of this circumference.
3. Plot the Euclidean line that passes through one of the intersection points of the former circumference with the boundary line and for D.
4. Consider the intersection between the line plotted in the first step and the line plotted in the third step. We name B this intersection. It can be proved (Workshop of Hyperbolic Geometry.pdf, in Catalan) that this intersection exists and it is a point of the hyperbolic circumference that we want to determine.
5. To construct the other point of the hyperbolic circumference that belongs to the line plotted in the first step, we used the homothety with center in the foot of the perpendicular in first step, O and reason CO·CO/BO. B is transformated into a point E that it is at the same hyperbolic distance from the center than the point D and B to be homotheties isometries for the half-plane model.
6. 7. Construct the Euclidean midpoint, M, of the Euclidean segment ending in B and C.
8. Plot the Euclidean circumference with center M and radius the segment from M  to D. This circumference is the hyperbolic circumference we wanted.

To construct the hyperbolic circumference we used that the homotheties are isometries for the half-plane model, so they  preserve distances. Moreover, in Workshop of Hyperbolic Geometry.pdf (in Catalan) we proved that the third axiom is true using almost the same construction we have followed here. The only difference is that there we have the point in the same perpendicular to the boundary line as the center and here we suppose that it is not true. So, the first thing we did is to transport the point in the same perpendicular, keeping the distance to the center. Then, we are in the same hypothesis.

List of tools
Hyperbolic geometry