 Parallel lines

In Hyperbolic Geometry, there are infinite lines that go through an exterior point to a line and do not cut this line. Of all these lines we considered that two are parallels with the given line.

In order to use this tool, it will be necessary to give three points. The two first points we will consider that belong to the line and the third will be the exterior point to the line.
We will suppose that the two first points are not in the same perpendicular Euclidean line.
So, in this situation we will be able to construct the two parallel lines easily.
1. Plot the hyperbolic line that goes through the two first given points.
2. Consider the two intersection points between this circumference and the boundary line.
3. Plot the hyperbolic line that goes through the exterior point and one of the former intersections. This is one of the two parallel lines.
4. Plot the hyperbolic line that goes through the exterior point and the other intersection. This hyperbolic line is the other one.

This construction is correct whenever the exterior point does not belong to any perpendicular line to the boundary line going through one of the intersection points. In this case we will not be able to plot the hyperbolic line that goes through these two points. In the other cases we obtain the parallel lines because they fulfill that they cut the given line in a point of the boundary line and this is the definition that we consider of parallel lines.

List of tools
Hyperbolic geometry