## Investigation: Does the 1st De Villiers point exist in elliptic (spherical) geometry?

Consider the sketch below where *G* is the incentre of the spherical triangle *ABC*, and *D*, *E* and *F* are the respective incentres of *ABG*, *BCG* and *CAG*. Drag any of the vertices 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, 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.