This custom tool can be used both as a tool and as the command Pline[ <Point>, <Point> ] in the Input bar. In addition, the custom tool can be saved to your computer as GGT file and then imported into other GeoGebra files. To do this, select Tools/Manage Tools, then select Save As and press Save. Then open a new GeoGebra file and use File/Open to import the GGT file. Be sure to do this prior to beginning the hyperbolic circle construction, since this approach hides all of the details of construction of the hyperbolic line tool and allows you to focus on constructing the hyperbolic circle. This works since all of the details of the construction are contained in the GGT file and they are not displayed in the Algebra window.
  1. Hyperbolic circle tool
Hyperbolic circles in the Poincaré disk model are also Euclidean circles but in general the hyperbolic center is not the same as the Euclidean center. The key to constructing a hyperbolic circle with center at points A and passing through point B is to locate its Euclidean center C and then construct a Euclidean circle with center C passing through B. The construction of the hyperbolic circle is broken into three cases depending on whether or not A = O and whether or not the A and B are collinear with O.
  1. Hyperbolic center at the origin
If A = O, then by Theorem 4 the Euclidean center of the circle is also at O and the hyperbolic circle can be constructed using a Euclidean circle centered at O and passing through B . Begin the construction by creating γ and two points inside γ.