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.