Hyperbolic angle

Given three ordered points, this tool will measure the hyperbolic angle considering that the second given point is the vertice of the angle. To measure the angle, it is necessary to consider the Euclidean tangent line to the hyperbolic lines that forms the angle.

1. Plot the hyperbolic line that joins the two first points.
2. Plot the euclidean line, r, tangent to the former hyperbolic line going through the second point. 
   
3. Plot the hyperbolic line that joins the two last points.
4. Plot the euclidean line, s, tangent to the former hyperbolic line going through the second point.
   To choose which of the four Euclidean angles is the Euclidean angle that we have to measure we can proceed in the same way as in the hyperbolic bisector (steps 5-10) where, in fact, we determined a hyperbolic angle.
5. Plot the perpendicular line to r  that goes through the first point.
6. Consider the intersection of the former line with r.
7. Plot the Euclidean ray that starts in the second given point and passes through the former intersection. This ray is one of those that will form the Euclidean angle and the intersection will be one of the ending points of the ray.
8. Plot the perpendicular line to s that goes through the third point.
9. Consider the intersection of the former line with s.
10. Plot the Euclidean ray that starts in the second given point and passes through the former intersection. This ray is the other that will form the Euclidean angle and the intersection will be one of the ending points of the ray.
11. Calculate the Euclidean angle from the intersection point of the step (6), the second given point and the intersection point of  the step (9).

When we use this tool the construction keeps the same elements as before and writes the value of the amplitude of the angle.

List of tools
Hyperbolic geometry