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?

Please enable Java for an interactive construction (with Cinderella). Below is a static sketch if the Java applet doesn't work.

Elliptic De Villiers2 Test

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.