## Investigation: Does the 2nd De Villiers point exist in hyperbolic geometry?

Consider the sketch below where *D*, *E* and *F* are the excentres of the hyperbolic triangle *ABC* on the Poincare' disk, and *G*, *H* and *I* are the respective incentres of *ABD*, *BCE* and *CAF*. Drag any of the vertices of the hyperbolic triangle to investigate whether *AE*, *BF* and *CD* are always concurrent. What do you notice? Can you prove (or refute) your observation?

Created by Michael de Villiers, 24 January 2011 with Cinderella. A free version of Cinderella 1.4 - the only dynamic geometry software that can also do elliptic (spherical) geometry and hyperbolic geometry - can be downloaded from Download Cinderella 1.4.