## Investigation: Does the 2nd De Villiers point exist in elliptic (spherical) geometry?

Consider the sketch below where *H*, *K* and *L* are the excentres of the spherical triangle *ABC*, and *E*, *F* and *G* are the respective incentres of *ABH*, *BCK* and *CAL*. Drag any of the vertices to investigate whether *AF*, *BG* and *CE* are always concurrent. What do you notice? Can you prove (or refute) your observation?

Created by Michael de Villiers, 29 December 2010 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.